270 - Tim Maudlin & Jacob Barandes: The Indivisible Approach to Quantum Theory

29:02He says either it represents nothing or it represents something but not everything, or it represents everything in physical reality, right? I mean, those are your choices. Some people try to get away with nothing, like the quantum Bayesians who say those wave functions, they don't represent the physical world at all. They're about our beliefs. They, you know, I mean, you know, Chris Fuchs will say psi should be written on your forehead because it's a representation of your information or your knowledge or your beliefs or whatever. Okay. That's a view. The many worlds people say it represents everything.

29:41Okay. Absolutely. Everything in the physical world is one way or another coded up in the wave function. Then you have the Bohmians and others who say, well, it represents something, but not everything. It represents something non-local. There's more than that. There's some local stuff. This is the way of Bohm and the way of collapse theories that introduce a local ontology. Okay. Now, let me just say first, before I wanted to get to something else but the view that it represents nothing which sounds like what Jacob is proposing okay then you can clear me out after this but let's begin with with kind of intuitive reasons to think it represents something which is just the regular two slit kind of phenomenon where you get these interference patterns um we the way we know how to predict interference patterns is by waves something governed by a wave equation.

30:41Water waves do it. Electromagnetic waves do it. Because the Schrodinger equation, which is the standard way the wave function is supposed to evolve in time, is a wave equation, it naturally gives rise to interference effects, constructive and destructive interference, these interference bands. And so that's a reason why everybody I know thinks it represents something. OK, there's something in the world. It's true. The mathematical gadget is defined in this high dimensional space. And that's puzzling because it can't represent a field like an electromagnetic field, classical electromagnetic field, which is on physical space.

31:24So there's a puzzle there, but just the interference effects usually lead people to say, hey, something's going on here, right? Something's in contact with both slits, right? Something's interacting with both slits because the phenomena depend upon whether only one slit is open or the other slit is open or both slits are open. All right. So that's notice a kind of causal condition here. You're saying, look, the state of each slit is causally relevant to the effect. So there must be something that's inner that's interacting with both slits and then going on and forming this interference in kind of the usual water wavy kind of way.

32:01okay um if you know the main questions you have of any physical theory is what does it postulate and how does that behave right what is there and how does it behave what's the ontology and what's the dynamics this is a reason to believe some real part of the ontology corresponds to the wave function and that it evolves by a wave equation like the schrodinger equation um i think but again he can correct me jacob doesn't think that at a fundamental level that is what's going on that maybe you can introduce you can have talk in those terms but that's not fundamental behavior it's somehow emergent okay so this is just one thing now the other thing i wanted the main thing i want to talk about is this Markov process, Markov condition, because it's a very simple condition, but now we're getting into an entirely different field, which is causation, causal models, the analysis of causation.

33:03And I just want to very quickly run through this so people understand. It's a truism that correlation is not causation, right? Everybody knows you can have correlations between A and B, but there's no direct or even indirect causal relation between A and B. On the other hand, all our evidence for causation is correlations. That's the evidence we have. That's what we point to in the data to argue that there's causation is correlations. So even though you can't define causation in terms of correlation, you use it to test. And just to be clear, because this is standard literature, but it's not talked about in people talking about quantum theory so much, but just standard literature.

33:52So take a correlation that exists, that for sure is there in the data, between people who have ashtrays in their houses and people who get lung cancer. Okay? So what do I mean a correlation? I mean, okay, take the percentage of people, and these are just percentages, these are just statistics, take the percentage of people who have ashtrays in their houses that get lung cancer, the percentage of people without ashtrays in their houses who get lung cancer, they're different. And there's a positive correlation because if you've got the ashtrays, your higher percentage gets lung cancer. but nobody thinks that ashtrays cause cancer and nobody thinks that cancer causes ashtrays, right?

34:39So that correlation we think is explainable in terms of a common cause, in this case smoking, right? Because some people smoke, they're both more likely or more of them have ashtrays around and a higher percentage get lung cancer. How do we test that? Well, one thing we do is we say, all right, let's just check whether if we condition on smoking. So we add, we had two things to begin with, which is ashtrays and cancer. Now we're going to put in smoking as a third variable and we're going to condition on it. We're going to say, okay, let's just take the smokers and now see how many get cancer and whether adding the information about the ashtrays makes any further difference to the statistics.

35:31And if it doesn't, which it doesn't, or at least, you know, the first order doesn't, then we say, ah, smoking screens off these two variables, these variables that are correlated, the ashtrays and the cancer. Once we condition on the smoking, the dependency on the ashtrays goes away statistically. It just doesn't exist anymore. So that's called a screening off condition. It's very important when you're doing causal analysis because screening off gives you reasonable clues about what the causal structure is. Okay. Now, um, the Markov condition that, that Jacob is very boldly suggesting we consider abandoning, which has been part of all physics up until now, and not just deterministic, but also indeterministic physics, right?

36:33You might postulate that things aren't deterministic, but still you would normally postulate the Markov condition. What does that say? Well, suppose, here's a correlation we have. We know that people who get shingles in old age had chicken pox when they were young, right? And so you say, ah, well, there's certainly a correlation there. In fact, if you never had chicken pox, you'd never get shingles. And if you've had it, there's a certain percentage of people get shingles. Now, suppose we have two identical twins and one of them gets chicken pox in their youth and the other doesn't. And then the doctors say, well, the one who got chicken pox, she's completely recovered all the traces are gone the two twins are back to being biologically and medically identical at say age 10 then normally you would not expect for the shingles to show up later for the one and not the other right that there'd be more percentage at why because you think well if the shingle pox at age six, if the chicken pox at age six was one of the causes of shingles later, that had to be transmitted through time somehow in their body, right, with some antibodies or virus or something hanging out inside the body from the early age to the late age.

37:59And so you'd say if we could condition, statistically condition, on the entire state in between the earlier one and the later one, that should screen off in earlier information from later events. It's just the same idea that that causation from the past to the future only occurs because it's carried continuously in, you know, continuously between the past and the future by something physical. Something physical is carrying it on. And so that's one problem you have in the causal monitoring literature is just that causal models, which tend to be just graphs, directed graphs, get very complex. the number of possible ones blow up exponentially.

38:47And so all the time, researchers are looking for constraints they can put on so they don't have to search through all of them. And those constraints often come from beliefs about what causes what because of time order and so on. But also the Markov condition is just assumed, right? You think it couldn't be that the earlier episode of chickenpox just reaches out across time from their early age to their seniorhood and induces shingles, right? It had to be by some intermediate continuous process carrying that information forward. Now, of course, if you give up on that condition, you have a whole bunch more possibilities.

39:32I mean, you're getting rid of a constraint. And whenever you get rid of a constraint, you open up things. but that's a pretty radical suggestion right it's what we might call temporal action at a distance people have in people know in quantum mechanics there's a lot of worries about spatial spooky action at a distance and Einstein didn't like that and a lot of people didn't like it and wouldn't accept it and shouldn't have accepted it until Bell gave us very strong reasons to think you can't get away from it. One can imagine that maybe we want to give up this Markov condition and say earlier facts can have an influence on later facts without having any influence on the intermediate facts in time.

40:23But I would think you'd really need to be pushed into that in a very strong way, right? And even stronger than some aesthetic preferences or something like that. I mean, it's a really big deal. And so I think, as I understand it, I just believe the wave function really does represent something physical. Yes, not something local, not something that's in regular space, but it's there. And it evolves continuously through time, maybe deterministically, maybe not. I'm certainly willing to imagine both. I'm happy with stochastic processes, but continuously through time. And that that's a deep part of the explanation of how earlier events can cause, be causally relevant to later events through this continuous process.

41:19My understanding, which Jacob can now correct, is that he doesn't, he's going to go for a theory where that isn't true anymore. and um i'm a bit puzzled about what the because it's such a big step right it's such a huge step i'm not even quite sure how you represent laws without a continuous evolution i mean there's all kinds of issues but i just wanted to say when we talk about the markov condition which is a fancy word probably a lot of people never heard of it's just the idea that earlier events can only be causally relevant to later events via intermediate events. Right. So that's just the thing I wanted to get clear about.

42:06Thanks, Tim. Markov, just for the record, was a person and introduced a certain kind of matrix in, I think it was 1906. And that's how the name Markov has gotten into this whole story about probabilities and stochastic development and so forth. So I want to, before I answer Tim's question, I want to clarify something that is a subtlety that was certainly difficult for me in working on this project. And this is the problem that there's a lot of terminological collision that one finds between different topical areas. So, and Tim's question was perfectly phrased. There's no question there. But I do think that a lot of people who may be listening to this might benefit a little bit from just disentangling some terminology.

43:05Words like Markov or non-Markovian, words like latent, we'll talk about latent and hidden. We'll even talk about the word model and even the pictures that are sometimes used. Words like indivisible. These are words that have conceptually different meanings in the theory of causal modeling, the theory of stochastic processes, and in quantum theory. What makes this extra confusing is even some of the diagrams people draw, these directed diagrams, often look very similar in the theory of causal modeling and the theory of stochastic processes. I can tell you this was an endless source of confusion for me trying to understand what had been done in the research literature.

43:59A causal model can have what's called the Markov condition or the causal Markov condition, and it can be related to a stochastic process that is not Markovian. And a Markovian stochastic process can be related to a causal model that fails to have the causal Markov condition. So this is all going to be, and there's just, there's no way around just disentangling this terminology. So I want to be very precise. One of the things that I really enjoy about talking with Tim is that Tim appreciates the importance of being very precise about things. So let me first just quickly talk about the causal model question.

44:38So if you pick up a book like Judea Pearl's classic 2000 book on causality, he introduces his extensive, beautiful theory. it's practically very useful and he talks about causal models so causal models are relationships between i guess features of the world they're often represented by variables and these variables are connected together with causal relationships directed causal relationships from causes to their effects and often these these causal models are depicted in graphical form one indicates the variables the features of the world with what are called nodes and a node is like a circle or a bubble and then the nodes are connected by line segments with arrows on them they're called directed edges and so you get what's called a directed graph often people will impose further that this is a directed acyclic graph which just means the arrows don't come all the way around, which would be like a snake eating its tail or some kind of weird causal loop.

45:51Now, to be clear, also, you can consider models where there are loops if they're feedback loops, but then you have to, you know, it's complicated. If you're thinking about the variables, you know, looking forward in time, for example, usually you want the arrows to not go loops. Can I just say a word there again so people don't get confused? The kinds of feedback loops you're talking about are between generic things, not particular things. So, you know, an A-type event can then affect the B-type situation, which then feeds back to an A-type event. But when you're talking about particular events, if you put in cycles, you got retrocausation.

46:30You're saying the future can affect the present, blah, blah. That's an entirely different deal, right, than the kinds of feedback groups you're talking about, which could happen in radios or amplifiers and microphones or whatever. Yeah. So for example, in the chicken and egg problem, chicken-ness and egg-ness are in a feedback loop, but any particular chicken is not in a feedback loop with the egg, right? Chicken lays an egg, the egg doesn't lay the chicken. So yes, like I said, you can get very precise about all of this. Okay. So we have this picture. We have nodes that represent the variables, and we've got these edges that represent causal relationships.

47:10All of the arrows that directly go into one node from other nodes define what is called parenthood in these models. We talk about all the nodes that directly have arrows into a variable. We call those the parents of that node. And the causal Markov condition precisely stated says that once you've conditioned, once you've fixed what's going on with all the parents that directly lead into a node, then that node cannot have any statistical dependence on anything else except maybe its descendants, the things that it has arrows and do. This is called the Kozomarkov condition. And if your causal model lacks the Kozomarkov condition, meaning that there are extra correlations or statistical dependences that are not screened off by fixing what's going on with the parents, then the usual attitude is to look at this model and say, we're missing something.

48:14There's some latent or hidden structure that we're not observing. And if you pick up Judea Pearl's book, you look in section 2.9.1, he talks a lot about this importance of, if it's missing the causal mark of condition, usually you want to go look for what's missing. And the thing that's missing may not be readily observable, but it's probably something that has a physical significance. You know, some connection that we didn't see. And if we knew what it was and we could put it in the model properly, then the model would obey the Kosomarkov condition. So that's called the Kosomarkov condition. Now, separately, we'll talk a little bit about the theory of stochastic processes.

48:50The theory of stochastic processes is a statement about some kind of system or object and how it changes, how it changes with time. And there's a formal way you can talk about stochastic processes using time index families of random variables. We can get very technical, but I want to just use it for now a very pedestrian view. It just describes how things change with time. Some rule for how things change with time. A list of things, numbers if you want, over time is called a time series. And if we allow the thing at each time to be a variable, then we call it a stochastic process. And for a stochastic process, it may be the case that configurations of the system at previous times play some role or part of the law that fixes either the next state of the system or fixes what its probabilities are going to be.

49:50if the next state of the system, the next configuration of the system, or a suitable probability distribution for the next configuration of the system, if the laws prescribe that they are fixed or determined solely by the previous configuration system, and it gets a little complicated if you've got continuous time, but we can talk about the subtleties there if you want. But basically, if you only need the previous time for the law to then fix either the next configuration or at least a probability distribution, we say that this is a stochastic process that obeys. You could call it dynamical Markovianity if you want to distinguish that from the causal Markov condition of the causal model.

50:30If that's not the case, if it fails to be the case, then we say it's a non-Markovian stochastic process. And you can construct stochastic processes that are not Markovian but status for the causal Markov condition. An easy way to see this is just take the parent nodes to be whatever previous configurations you need for your system. And once you have enough of those previous configurations and you know them, that will determine the next configuration of the system. It'll screen off everything else. And then it satisfies the causal Markov condition. So Judea Pearl might look at this and say, no problem.

51:03I mean, if we didn't have a causal model that obeyed the causal Markov condition, but we could include a few past states of the system, now we have a causal Markov condition. Now with these additional variables in the past, we stick them in the causal model. and now the causal model satisfies the causal Markov condition. The parent nodes representing multiple previous configuration system are enough to screen off any other influences. And on the other hand, you could have a Markov process where you only need the previous state and time for the law to then determine the next state or the probability distribution.

51:33But it could be that each configuration is some large collection of variables that have weird correlations between them that aren't screened in any way. And if you try to write that as a causal model, it might not obey the causal Markov condition. So I know this sounds very technical, but it's really important to make clear that there are these conceptual differences between things. When I talk about the status of the Markov condition, I'm talking about the status of the Markov condition at the level of a stochastic process, not necessarily the level of a causal model. However, there is a very interesting relationship between the two.

52:06So it's not that I'm saying let's take a causal model, some beautiful, elegant causal model that satisfied the causal Markov condition. And by the way, the wave function or the thing it refers to the quantum state. And I should say, by the way, just to be clear, when I said quantum state, I did mean the mathematical quantum state. And Tim is right. I was saying quantum state to refer broadly to whatever ingredient either in the Hilbert space, and it could be a what's called a density operator or a rank one density operator or a single vector, or the quantum state could be what's called an element, a positive normed linear functional in the dual space to a C-star algebra.

52:42There's many ways to talk about the mathematical quantum state that get more and more abstract. But if you want to think of the quantum state as its physical thing, the thing, the referent, the physical referent, the thing being referred to by the mathematical quantum state of the wave function. So, you know, you might imagine, oh, so I took some causal model that satisfied the causal Markov condition. It had the quantum state in there. It's all great. And then I, Jacob, came along and just ripped the quantum state out of it. Now the model doesn't have the Cosimarco condition. Now it looks ugly and complicated, and Jadier Perl would say don't do that.

53:15That's actually not what I'm doing. So to explain what I'm doing, it would be helpful to tell a few stories. But this would be a good point to pause for just a moment because I want to make sure that... Yeah, actually. So let me also make a kind of conceptual comment, for those of you playing along at home in your own minds. Yeah, there's causal modeling literature, especially linear causal models, which are very useful for various purposes. I was using that literature to illustrate screening off and the relevance of screening off to causation. But I'm thinking in terms of, I mean, I have this very impractical but nice chance, like all physicists, to say, well, ultimately, it's like you ask, what were the causes of some event?

54:09ultimately everything in the past light cone or you know i mean the the entire state of the universe that's the cause right because why because i kind of have to specify all of that to give me something that my equations have traction on my physical equations have traction on now from a practical point of view if you say what was the cause of the accident you say well the entire previous state of the universe you know that's not going to help you much when you go and have to defend yourself in court. And so we, of course, think of the world as broken down into more localized things that have much more particularized causal relations between them that we represent with these graphs, with nodes, these directed graphs in the causal modeling literature.

54:54But those are clearly simplifications. I mean, I take it we all agree. Those are simplifications. they're illustrative for conceptual purposes what we really want is to get down to the physics of it and that may well be global um where the you know the causal graph of it is very boring it's just you know that's the entire state creates the entire state creates the entire state or whatever but you can still ask there but gee does it screen off right does does a later state get screened off from the further pass by a state between them or not. So yeah, I am thinking in terms of the stochastic process notion that you have in mind.

55:41I illustrated it because I think there are people who don't follow physics who may be familiar with the causal modeling literature, and there are kind of conceptual connections that are close enough to give you a feeling for what's going on. That's all. I have a question actually for you, Tim, about this. So in my experience, analytic philosophers, philosophers of science, philosophers of physics seem to be very familiar with causal modeling. In fact, I would argue, and I think I argued in a previous conversation with Robinson, that one of the most fruitful relationships between science and philosophy is between statisticians and people who work in, statisticians and philosophers, but especially statisticians who work in things like cause and modeling.

56:26I mean, you read Judea Pearl's book, for example, and it's filled with lots of discussion of actual philosophers, you know, Nancy Cartwright, David Papineau, and various people. But, Tim, in your experience, do philosophers have a lot of familiarity with the theory of stochastic processes? That has not been my experience. Okay. All right. So this is a little inside baseball, but just so people know, my dissertation advisor was Clark Glemore. Clark was doing work on causal models and causal search. Wonderful book. This actually, Pearl learned a bunch of the stuff he writes about from Clark and Peter Shinas and his students.

57:06So I'm familiar from there. But when I was in graduate school, Ken Schaffner was interested in medical methodology and in causal models in medicine. So, of course, you got used to it there. and Nancy Cartwright, sure, a lot of people who are not doing physics but are looking at the special sciences where you have these simplifications are very familiar with linear causal graphs and causal modeling there. As far as stochastic processes go, okay, when I got out of graduate school and came to Rutgers, I met Shelley Goldstein, who had been working on Nelson's theory, and so he was awash in stochastic processes.

57:46So I got kind of familiar with stochastic processes myself very early. I guess the people who work on pilot wave theory are kind of aware that you can do stochastic versions like Nelson's stochastic mechanics. So I think a lot of them know at least the basics of it, but that may be a little niche. Yeah. I mean, that would be more niche than knowing about causal model. Right. That's been my experience that it's sort of, you find And it's sparsely distributed. I should say that the connection between stochastic processes in physics is also very complicated. I think I may have mentioned elsewhere, again, maybe Robinson, you and I, when we talked, that in Hugh Everett's thesis, he talks about some work.

58:31This thesis he's writing in 1956, 1957, he talks in his thesis about Fritz Bopp working on a stochastic process interpretation of quantum mechanics. that was an early forerunner to what we would now call Nelsonian stochastic mechanics. These were all based on using things like Brownian motion, Wiener processes, inherently Markovian type stochastic processes. When I speak to physicists and I say the word stochastic process, or really myself, because I was also trained in philosophy, but also in physics, when I thought of stochastic processes, I always thought of Markov chains, Markov processes.

59:06The idea that a stochastic process was inherently Markov was just always taken for granted in my experience among many physicists. Now, I'm sure there'll be people watching this saying, I don't take it for granted. It's true. There are absolutely some situations people don't take it for granted. But broadly speaking, in my experience, it was. So you get kind of a tunnel vision and it's very hard to see what else could be there. Let me quickly address a point that Tim very, very sagely made. Which is that when we're doing causal modeling for macroscopic, you know, everyday stuff, medical stuff, cigarette smoking and so forth, the causal Markov condition is extremely important.

59:47It's very important for causal discovery. It's very important for how we think about what's going on ultimately. We, of course, have very, very little experience with the structure of causation when phrased solely and directly the level of the atoms of nature. and judea pearl i should i should add is is quite up front about this in fact he even introduces a word he says maybe at the level of quantum mechanics we need to replace causality with causality um and he doesn't know what that and he says in in the section on latent structures he's like i don't know if this generalizes all the way down to the microphysical regime of quantum mechanics, but certainly it's got to somehow emerge from macroscopic stuff.

1:00:32So what I'll tell you is that although I don't take a causal model and just rip the causal Markov condition out of it, the causal models that will emerge at the microphysical level, the level of the atoms, I say atoms here metaphorically, the level of whatever the myriological smallest components are, will not initially have a causal Markov condition, but it emerges in the macroscopic limit, it, which is when we need it. But that's farther downstream. Let me start with a story. It's one of my favorite papers. And I know Tim knows this paper. It's actually one of the first papers that used the phrase temporal action at a distance, not the first paper.

1:01:11I think the first paper I was able to find was a paper by Arthur Papp, I think from 1952. But the paper I'm referring to is a paper by Sidney Shoemaker, 1969. And the paper is called Time Without Change. And Shoemaker is trying to answer a question, a very old question that goes back to antiquity, is our expanses of time, do they necessarily correspond to changes in physical ontology? Does it make sense to say that you could have some expanse or duration of time, but nothing is different from one part of that expanse of time to the other, the level of the physical ontology. And the, I guess, gut impulse one might have is, of course not.

1:02:02How could it make any sense? And then Schumacher asks, could we have any good empirical reason to think that that could be possible? And you'd say, well, even if you thought metaphysically or, and here I'm using metaphysically in the colloquial sense, not, I just mean in the sense of, you know, beyond, maybe beyond empirical, you know, whatever. But how could we ever have any empirical evidence that there was a stretch of time in which nothing changed? That's impossible. And he says, well, how about the following scenario? And I'm going to change some of the numbers a little bit because I like these numbers a little better than the numbers that Shoemaker used.

1:02:36I think they're a little more evocative for a variety of reasons. But roughly speaking, this is the picture that Shoemaker describes. Because I think people listening might go, this is ridiculous. How could you ever have empirical evidence? But just hang on. Shoemaker says, consider a hypothetical different world or universe, just a hypothetical other universe. And this universe is some expansive space, some featureless, enormous, infinite, whatever, no boundary, just huge, empty space. And the only thing in this universe is three civilizations that are separated from each other in space. They're all like in some compact region.

1:03:14There's a compact region for one of them called that Civilization A. And some large distance away, there's a compact region where Civilization B lives. And then some large distance away from the other two, there's another compact region where civilization sea lives. And there's it's empty space in between. Each of them is like a little little island in space with, you know, maybe it's a planet or something like that. But there's basically nothing between them at all. and these civilizations they know about each other you know they're not so far apart you know they can look in telescopes, they can see each other.

1:03:46And well, civilizations B and C, they notice something rather odd. They notice, and I'll just say for the sake of simplicity that, you know, I guess they're all orbiting a star or whatever, their own star, they have a notion of the year. But civilizations B and C look at civilization A and they notice that every seven years, civilization a just freezes like they're doing their thing and people are talking and then everybody stops everything stops there's no motion no activity just everything has come to a halt brain activity perception of everything is just frozen and it remains frozen for one year and then at the end of the year everything just starts moving again and none of the inhabitants have any memory or perception that there was a gap in time.

1:04:37And civilizations B and C notice this happens every seven years. They're like, huh, that's funny. They don't seem to be any worse for the trouble. So, okay, I guess that's just the way things are. Civilizations A and C look at civilization B and they see that every 11th year, the same thing happens to civilization B. It totally freezes for one year and then it unfreezes. Okay, that's weird. And I say freeze, I don't mean it gets cold. It's just everything stops and then it starts again. So no one is hurt. There's no harm to anybody. And again, all the inhabitants of civilization B have no perception that there was a gap for them.

1:05:10It was just like they jumped over that that year with no perception of the time in between. And then, of course, obviously, you see where the story is going. Civilizations A and B, they notice that civilization C has a similar pattern, only it happens to them every 13th year. Every 13th year, they go through a year. They're frozen. And this somehow reminds me of cicadas. I don't know why, because, you know, cicadas, you may know they're on. There's a certain kind of cicada that that only comes out every 13 or every 17 years. And it's one of the rare cases where prime numbers show up in the natural world.

1:05:40And there are all kinds of conjectures about why this is, you know, trying not to sync up with other with predators, maybe, or not trying to over. There's all these theories about it. But we just noticed that prime numbers show up in this case. So somehow this story reminds me of cicadas. OK, but then there's an interesting question you can ask. if you so 7 11 and 13 are prime numbers that means their least common multiple is the product of the three and you can do the math the reason i pick these numbers is i wanted a very big number but also it's very nice 7 times 11 times 13 is a thousand one it's nice it's a really nice number and and so what are they to make of what should happen on this you know on the thousandth first year i mean if you just look at this you'll go well i i guess the thousandth first year they should all be frozen for a year but of course none of them have any experience of that they're all frozen none of them have any memories of that uh so do they have good empirical reason to believe that they are in fact all freezing on the thousandth first year or should they have a more complicated model of the world where every thousandth first year the usual rules don't apply and there is no freezing.

1:06:53Like what would be, if you were an inhabitant of this civilization, what would be your guess? You'd probably say, you know, it does seem kind of plausible that we are in fact all of us freezing. But of course, if we all freeze for one year and we're the only things in the universe, the only physical ontology in the universe, then are we committed to the idea that there's a duration of a year in which there is no change to any of the physical ontology? and i just think this is such a beautiful example because who would have thought there could be any good reason to think that this could in fact be going on yeah i mean can i just make a comment here um some of this goes back of course to aristotle who says time is the measure of change right that's what i mean that's right but it's also the case that time is the measure of stasis right if something's not moving you still say there's a certain period of time It has to be not moving for a period of time or else it's not stationary.

1:07:49And so time is also the measure of stasis. Now, you might say, yeah, but if all the clocks have stopped, we could never know that things had frozen. And now, and this actually is a bit relevant to Heisenberg and so on. You think when Heisenberg, 26, were about the time of logical empiricism, there was in the philosophical world, in the Vienna circle, a very strong push to this idea that you need to reduce all meaningful language to observation language, right, to the protocol language for philosophers who know this stuff. And that you just couldn't even have meaningful language that went beyond what was observable.

1:08:35And people thought this was the moral of relativity, actually. Don't talk about space and time as things in themselves. Talk about instruments, clocks, and, you know, how we use them and, you know, this kind of instrumentalism. Um, there's this wonderful story that Heisenberg himself tells that he gave this talk on matrix mechanics in 1926, University of Berlin. And Einstein came and he was so interested that he walked back with Heisenberg. and and Einstein says Heisenberg you can't be serious you said that you know you you talk about electrons but and and being in atoms in their motions but then you say the theory must rest must be formulated in terms of what's observable he says that doesn't that doesn't make any sense right there's a famous line here right and uh it's the it he says it's the it's the theory that tells you what you can observe, right?

1:09:37You have to understand the theory and then analyze it to see what the observations will be like according to the theory, right? And Heisenberg says, but Einstein, isn't this what you taught us with relativity is that we have to think? And Einstein says something like, well, maybe I said stuff like that, but it's nonsense all the less, you know, nonetheless, it's the theory that tells you what you can observe. And this then chimes, you have to jump from the Vienna circle up to Quine and conformational holism and this idea that, look, yeah, we could have evidence. I mean, I don't really care whether the.

1:10:15Yes, we were frozen for a year or there was a weird glitch that it changed all the rules. Which one is more plausible? The point is, they're both things you want to consider. I personally think that we were frozen for a year would be the better. But OK, I don't want to argue about that. But the point is that you have to take the entire situation and the entire theory and all of the observations and all of the explanations into account to make judgments of scientific plausibility. And what Shoemaker gives you a case where even the empirical data together with very plausible assumptions would lead you to the conclusion that everything froze for a while, right?

1:10:57That time went on with the physical state unchanging. Now, to come back to your first point, this would require a kind of temporal action at a distance because somehow after a year, I mean, why do things start up again? Right. There can't be any immediate preceding cause because everything was frozen. Right. So you would need a kind of temporal action at a distance. And I'm not saying it's not inconceivable. It's not self-contradictory. It's not a feature of any physics we've had. And it doesn't seem to be a feature of the world we're familiar with. anyway. Yeah, it's great. And let me just say for people who'd like to read this story, which is a fantastic story, it's in Physics and Beyond, Heisenberg's book, his sort of memoir book about his life.

1:11:47I highly recommend it. And the details of that engagement between him and Einstein are really very interesting. Yeah. So, Tim, exactly. You're exactly right. Shoemaker talks about temporal action at distance because in a universe like this, it's the most natural way to describe what's going on. If you ask, does the configuration of this universe one second before an unfreezing, is that enough to predict that it will unfreeze? The answer is apparently no, because every second during the thousand first year that's frozen appears to be ontologically physically like every other second. And just from knowing the configuration at one of those seconds doesn't tell you that this is the last second.

1:12:36Somehow it would seem you'd need more information.

1:12:41Now, there is an alternative. You could think of this as, well, we wouldn't exactly call this a stochastic process because it's not really probabilistic. It's a trivial version of a stochastic process where there's no probabilities. but it's a dynamical process for sure. There is an alternative. An alternative is to turn into what's called a hidden Markov model. In a hidden Markov model, you introduce formally some additional variable. You introduce some variable to the model. This variable is sometimes called a latent variable or a hidden variable, and this is the other distinction I want to make between stochastic processes or dynamical processes and causal models.

1:13:25So although the same term is used, hidden or latent, it's not quite the same thing. In a causal model, like I said, you restore the causal Markov condition by introducing some latent variables, but you think those latent variables are probably representing something physical in the world. For a stochastic process, turning the process into a hidden Markov model means introducing some new variable. And with that variable, we can model the process as if it were Markov. The additional variable enhances the process to being a Markov process. But usually in the stochastic process literature, this hidden variable is not treated as some physical object or something like that at any given time.

1:14:05In fact, in a lot of models, it's explicitly taken to be a summary of the entire history, but just like stored up as one variable. This is what's more typical to view what we do with a Markov model. But let's just take the universe that we were describing, the Shoemaker universe, The cicada universe, if you want to call it that. Let's suppose that I formally augment this picture with a new variable. This new variable is not located anywhere in space. It's not local. It's some non-local. It maybe exists in some sense, but it's not fixed anywhere in space. And this variable is just like an incrementing, increasing variable in some sense.

1:14:50It simply increases with time. Now, if we have this variable available to us, then we can write the laws of this universe in a Markov way very easily. If you want to know what's going to happen to a civilization, you just consult this variable, right? You can't see the variable. The variable is not visible. It's unobservable. But it's not unobservable because it's small or complicated the way maybe a latent variable in a causal model might be. It's unobservable in principle. It's not something that has any kind of existence in the familiar sense that it is located anywhere that you could see it.

1:15:25But if you believe it exists, if there is a Schrodinger, let's say, in this universe who says, you know, I can turn this strange set of laws that don't appear to be Markov, I can make them Markov by introducing a new unobservable ingrained reality that isn't really located anywhere. Well, then you're going to get a Markov description, right? That variable at any given moment will determine what the universe is going to look like. There are a couple of interesting features of this variable that we've introduced, one of which is that it's not located anywhere, and the other is that it's not back-reacted upon by any of the physics.

1:16:02The lack of back-reaction is a really telling feature. So this extra variable we've introduced, this non-local clock, seems to be determining what's going on to the universe, but nothing going on in the universe in any way appears to be able to do anything to the clock. The clock does its own evolution and sort of reaches down in some sense to the universe if you want to imagine it's there. There's no back reaction. So the lack of back reaction is a very telling feature that you have as a hidden Markov model. So this is, and of course, One of the things that one will note in a theory like the Bohm theory, which has the wave function as a part of the physical ontology on some readings of the Bohm theory, and also has local ontology, the beables, the things that can be in simple models, the particle arrangements.

1:16:58You have both of these things. It's notable that the pilot wave doesn't live anywhere in physical space. It's non-local. And the particles don't back react on it. And this is very reminiscent of what goes on with the hidden Markov model. So before I go any further, this would be a good place maybe to pause. I want to, in a moment, present a very simple model that will motivate the indivisible theory. But let me just stop there for a second. Yeah, I mean, I think it's a good place because, you know, otherwise I'll just sit here and stew for a while. So let me get it off my chest. So just a couple of points.

1:17:35One is this idea that there's no back reaction. In the standard pilot wave picture, that's true. That is, you have the quantum state. It evolves always by Schrodinger evolution. and that's it you don't need there are these particles the particles are moving the particles are being piloted or guided by the state of it but there's no back reaction this has been a complaint forever especially from the many worlds people it violates newton's third law i'm not complaining to be clear yeah yeah okay i'm just yeah but but you think it's significant right you think it's Suggestive. Suggestive, perhaps.

1:18:18Okay, let's put it that way. And I just have a couple comments to make. One is, if you want a back reaction, you can put it in. I mean, I actually mentioned back reaction dynamics a while ago in a conversation I was having with David Albert and some other, he said, what's back reaction? I said, that's something I just made up, where you add an extra piece to the Schrodinger revolution that is a function of the configuration. Maybe it slowly kills off the empty branches, right? You can imagine doing it if you're that, you know, head up about it. You know, you could do it in a way that would probably be empirically unobservable.

1:18:54Do you want to just say what empty branches are, Tim? Yeah. In the pilot wave theory, it's just the branches that are parts of, as it were, associated with parts of configuration space that is not anywhere near where the configuration actually is. That's what that means. Because it's a term of art. I want to make sure people are listening. And in a collapse theory, the collapses will essentially kill them off. I mean, not completely because you use a Gaussian blah, blah, but reduces the amount, the magnitude, the amplitude of the wave function away from where the collapse center is. You could put a term in that kind of slowly evaporated off the parts of the wave function far from the actual configuration in a pilot wave theory if you were so inclined to.

1:19:43I can't, you know, I'm not concerned enough about this because I never thought that this analogical version of Newton's third law was anything I should pay attention to. I mean, I think, you know, as it were, in a way, the laws of physics influence the particles and the particles don't have a back reaction influencing the laws of physics, you know. So at this analogical level, I was never worried about that. But I do want to say you introduced this as I'm introducing maybe even self-consciously a fictitious object. And the way you described it made it sound pretty fictitious. Whereas when I started, I said, look, we need something that actually does evolve by a wave equation because we have actual phenomena that are interference phenomena.

1:20:37I mean, there's this comment that Bell makes about the two-slit experiment where he says, is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear from the interference that the particle is guided by a wave? Now, that wasn't to make it Markov or anything else. It's to say, we know the kinds of physical systems from classical physics that display interference phenomena. And they're governed by wave equations, and the wave equations are Markov processes. And so if you just go that way, you're going to get a Markov process. It's going to be written down as a continuous differential equation in time, which is kind of what we want and what we've always had.

1:21:24So I think it would be misleading to suggest that the motivation of Schrodinger, a pseudo-Schrodinger, was just, gosh, I really want Markovianity. I better put this new thing in to maintain it. But, you know, the straightforward motivation was, gosh, I've got an interference process. I better have something governed by a wave equation. Now, it turns out it has to be it can't just be in physical space. It has to be in this higher dimensional space. And that's connected with entanglement. And then that eventually with nonlocality. Again, this isn't just on a whim. I mean, you know, this stuff is being introduced for really very good empirical reasons.

1:22:05So I think if someone listened to this, the account you gave of how this new thing gets in, they'd say, gosh, somebody really is so attached to Markovianity that they're willing to postulate whatever to maintain it. But I don't think that corresponds to the history or certainly not a rational reconstruction. Of course, I'll just say one final word. When we rationally reconstruct quantum mechanics, we often start with the two-slit experiment. They didn't, right? They were talking about spectra and light coming off of atoms, and that leads you to matrices, and that leads you in a certain direction.

1:22:47if you start with two-slit interference, it leads you in another direction. But I think given where we are, starting with the two-slit thing, gives you a smooth story, plausible story about why one would not only do this, but hard to see how you could do anything else. So let me just say a couple of things. The first is, of course, in the Shoemaker universe, our uh uh parallel world schrodinger could certainly introduce some back reactions on the clock variable i mean that that would certainly be fine if you have some problem with the clock variable you can deform it lots of ways there are a number of other telling features that i want to call some attention to one is that the clock variable we've introduced is not just abstract and non-local and doesn't get back reacted on on the simplest on the simplest Markovian model, but it's also highly malleable to redefinition.

1:23:48For example, we could make our clock variable periodic. We could say that every thousand one years it returns back to zero. It's like really a clock with, you know, a thousand one years and then it returns back to zero, a modular clock. There are much more radical things that we can do to it. And this will become much more apparent once we start considering generalizations of the Shoemaker universe in which there's probability. So I'm going to come back to that, to those points in a moment. The next thing I just want to point out is, Tim, you're completely right. Schrodinger didn't sit down and say, you know, we need to make a Markov model.

1:24:25I'm going to introduce a fictitious thing to make a Markov model. That's not what Schrodinger did. You're completely correct. You can read, go back and read the papers. They're beautiful. The first few were in German, But one of the seminal papers that Schrodinger wrote in 26 was in English because he spoke and wrote in English as well. It's a beautifully written paper. It's where he talks about the undulatory theory of mechanics. I think undulatory is a much better word than wave. It's such a lovely word. And he makes clear that his motivation is coming from the Hamilton-Jacobe formulation of Newtonian mechanics.

1:25:02Now, I don't want to get too in the weeds here, but a couple of salient things are worth saying about this Hamilton-Jacobe formulation. In the Hamilton-Jacobe formulation of Newtonian mechanics, for at least some kinds of Newtonian systems, systems where we can represent the forces in terms of what are called potentials, there's a whole story there, there's a way to rewrite the laws of Newtonian mechanics. the laws of Newtonian mechanics take the form of a kind of not quite Markovian set of laws. They're second order in time. And second order in time laws are slightly not Markovian. You need to know the configuration of all the particles now and like slightly earlier, and the slightly is in the sense of a calculus limit in order to get the configurations that are coming up.

1:25:51Now, there are various ways to rewrite this story to make it look Markov. There's the Hamiltonian formulation, which looks more Markov. We can just take velocities to be fundamental aspects of reality. There's various things we can do. The Hamilton-Jacobe formulation writes the laws in a Markov way, in a rather curious way. There's a long story to how you get here, and I can tell you all the details if people here would like, but the upshot is you end up with a couple of equations that are all explicitly first order in time, explicitly Markov. these equations describe the evolution of an abstract function that's not local it doesn't exist anywhere in space it's in fact defined on configuration space uh it is not back reacted upon by the particle and this function is called the hamilton's principal function or the hamilton jacobi function and you solve a partial differential equation that looks a lot like Technically, it's called an iconal equation.

1:26:58It's very similar to a wave equation, but it's a partial differential equation. And then the gradient, the mathematical gradient of this Hamilton-Jacobe function determines the momentum or the velocity of all the particles, which is like a guiding equation. And you could have imagined people writing this down and going, oh my gosh, maybe this suggests that the best way to think about Newtonian mechanics is that there's this new entity, this non-local entity that exists. It's ontological. It's this Hamel to Jacobi function, and it guides the particles around. Everything is now first order. We have a first order velocity guiding equation.

1:27:41I think most people from a Newtonian persuasion would have said, no, no, the fundamental physical picture is this second order dynamical equation. And really, we're just finding a very clever way to reformulate it to make it look explicitly Markov, to give this very elegant picture with guiding and first order velocity equations and so forth. And although Schrodinger didn't talk about, this was before Bohm, this was before even de Broglie, who had a somewhat similar-looking theory. No, it was after de Broglie. Look, this was 26. De Broglie's theory was 27. De Broglie's even got the idea in 23, presents it, I think, in 25.

1:28:18But the pilot wave model? Explicit lighting equations? Wow, I didn't know that. I thought he presented that after Schrodinger. So I know that DeBruy had his matter wave hypothesis that came first, and it was his thesis, and Einstein won a Nobel Prize for it, his 1924 thesis. He won the 1924 Nobel Prize. Look, I remember someone telling me it actually went by anyway. I wasn't aware of that. You could be right. I'm not a historian. but certainly shorting his writings don't betray that he was thinking about particles being guided or anything like that but he did use the the hamilton hamilton jacobi picture he took the hamilton jacobi function that satisfied this partial differential equation and he said well it satisfies a partial differential equation that looks like someone took a high frequency or iconal limit of a wave equation if that were the case what would be the wave equation and how would the Hamilton-Jacobe function fit into it?

1:29:13And what he did was he essentially exponentiated the Hamilton-Jacobe function. He regarded it as the phase of a wave. And in fact, he drew these beautiful pictures. You can look in the paper. He draws what I call the level curves, the Hamilton-Jacobe function, and he shows how they would change with time. And it looks like wave fronts. But this is all happening in Newtonian mechanics. He hasn't gone to the... So you're seeing wave-like stuff, not empirically, but at the level of the mathematical formalism already in the Newtonian case and this inspired him to propose maybe the hamlet jacoby function is the phase of some wave function introduces this trident symbol this trident symbol is is a wave-like function for which the hamlet jacoby function is its phase like the hamlet jacoby function it lives on configuration space it's not it's not um defined in any local sense and then it evolves according to the wave equation.

1:30:06And then by looking at the so-called normal modes of this equation, he was able to relate that to the energy values of atomic spectra. That's how he arrived at this. And so although he wasn't saying to himself, I need a Markov description, the way that he arrived at the wave function by building it, constructing it out of something that a Newtonian probably wouldn't have regarded as a part of the ontology is also, I think, in retrospect telling, even though it wasn't, I think, part of Schrodinger's thought process at the time. Right. So let me just make two quick comments about this, because people, I mean, you said something that's technically correct, and I just want to explain exactly what you said.

1:30:49Well, Newtonian mechanics is almost Markov, but in a way, not quite. And the reason for that, as probably everybody knows, is that in Newtonian mechanics, the initial conditions you need are both the initial positions and initial momenta or initial velocities of all the particles. So if you say, well, just give me the state of the entire universe at a moment, then you have to stop and say, well, what do you mean? What's true of it at a moment? Laplace versus Boscovich, right? Yeah, this is great. Yeah, exactly. You might say, well, certainly the positions are all what they are. And so So the configuration is all what it is at a moment.

1:31:31But what about the velocities? Are there instantaneous velocities? There's a big philosophical literature about the conceptual possibility of instantaneous velocities. Do those exist at a moment? They're pegged to a moment, right? As it were, if the velocity is changing smoothly, you can ascribe a velocity at a moment. But you might say metaphysically, you need more than a single moment for there to be a velocity, right? Right. So that's true because Newtonian mechanics is a second order theory in time. That is, the fundamental dynamical equation is about accelerations, which is the second time derivative.

1:32:11So you've sort of one way to think of it the way Boschkovich did, which is right. I need a little sandwich. I don't need I can't just take an instantaneous state. I need a little sandwich of time, but it can be as thin as I like. Right. I can make it as small as I like. That'll give me all the derivatives up to any number of derivatives. So if that's the difference between a really Markov and quasi Markov, that's true. But notice, and this is interesting, that in the pilot wave theory, you don't have that problem at all because the two fundamental equations, Schrodinger's equation and the guidance equation, are literally first order in time.

1:32:50They're just single derivatives with respect to time. So really in that theory, You can say, just give me the state at a time in terms of what we would call categorical facts at that time. If the wave function represents a categorical physical fact and the configuration represents a categorical physical fact. And that's all. That's it. Right. You know, any any information about what happened earlier, you'll be screened off. Right. It won't it won't give you any more predictive power to know anything more. Right. Um, so that, that actually suggests, at least if you're going in a pilot wave thinking that this is more, uh, Markovian, right?

1:33:36It's more friendly to deep Markovian-ness than even Newtonian mechanics is. The second point is everything you said about Hamilton-Jacobe theory is certainly correct. The motivation is a lot of the mathematics that was used. It was familiar to people. They had it to hand. um the thing i started with with the two slit experiment was the interference phenomena right that you can have both both constructive and more importantly destructive interference that just looks like wave behavior i mean in a serious way and and that that if you have a quantum state represented by the wave function evolving by a wave equation that you would get that kind of interference is what you'd expect.

1:34:22I don't know of anything in Newtonian mechanics that would be analogous to that. Right. So I can see why the Newtonians would say, yeah, this is some nice abstract mathematics. At the end of the day, it all boils down to F equals MA. But but trying to say, well, how do I account for these interference bands? That's just a straightforward physical problem that the tools they had available are just suggestive of a wave phenomenon. Yep. No, I'm so glad you brought that up. I don't know if you can see my chalkboard from behind me, but you may see that I have a double slit experiment on my chalkboard.

1:35:01This is definitely something we're going to talk about. So I agree with everything that Tim has said. It's not, if you try to make an analogy to Newtonian mechanics, you have at a mathematical level, and I made clear to say this, something that looks kind of wave-like. Like the Hamilton-Jacobe function looks like wave fronts. It has a distinctly wave-like nature. We would call it the geometric optics limit type, sort of iconal approximate wave-like nature. But in Newtonian mechanics, we don't see empirically signatures of wave-like behavior in experiments. Now, the double slit experiment has a long vintage, right?

1:35:38I mean, the original young double slit experiment was, and now I'm not an historian, but my understanding was that it was part of a debate over the nature of light, whether light was made of Newtonian classical particles, the corpuscles, the corpuscular theory of light, or that light was inherently a wave. And the Young double slit experiment involved sending light through two slits and seeing an interference pattern. And that helped cement the idea that one should think of light. Or really originally the thing that's sometimes called the Fresnel spot and sometimes called the Arago spot and sometimes called the somebody else spot, which you probably know the story.

1:36:17I mean, it's an amazing story because, yeah, Newton had a corpuscular theory of light in the optics. There was a contest that they announced for, you know, that they did back then. And who was it sent in the wave theory? And then, you know, I'm getting I can't get the players straight. One of the judges said, no, no, no, no, this is crazy because if this wave theory were true, you'd see this funny spot. If you had a source of light and then a circular barrier, there would be a bright spot at the middle of the shadow. And that's crazy. and then they did the experiment and found it, right? And they go, oh, I guess wave mechanics is correct.

1:37:02You know, I guess the wave theory of light is correct because this was a prediction of the wave theory that was thought to be an absurdity, but actually showed up. You know, it's a great story. I wish I could remember the details of all the players. I mean, to situate this, of course, this is centuries ago. The Young Doubles that experiment was not recent. There is a little bit, is it okay if I do a little bit of a tangent that I think people here will like? And Tim, I think you'll actually also really appreciate this. It's a little bit of a tangent. So you probably, Tim, I'm pretty sure you probably know this.

1:37:34I'm not sure very many people know. The first theoretical prediction that the sun, the sun's gravity could deflect light predated Einstein's theory of general relativity by over a century. in the newtonian corpuscular theory of light the idea is that light particles are very lightweight particles lightweight light now in the in the sense of mass right very very lightweight famously newtonian mechanics doesn't really know how to deal with massless things very well because then there's nothing that shows up in the in the newton second law but um but if you treat particles of light if you think that light is made of classical particles just little little classical particles flying around, they could fall under gravity the way that other particles could.

1:38:23And Cavendish and Saldner proposed that maybe this could predict that the sun's gravity could deflect starlight, that as the light from stars was streaming at us, if it came very close to the edge of the sun, on its way to us, the sun's gravitational field could bend its path just like anything could fall under the gravitational influence of the sun. And then when we looked at those stars, we would see them in the sky if we could maybe during a total eclipse when you could see the stars near the sun. We'd see the stars apparently coming from a slightly different location because their light had been diverted.

1:38:57And Sautner did a calculation of how big the effect would be. And he got a prediction. And rather famously, this calculation was done in 1801, I believe, remarkably long ago. And this prediction was almost exactly correct. It was off by a factor of two. Of two. Yeah, it's a big fact. It's an important factor, that factor of two. It wasn't off by a factor of 100. It wasn't, you know, it was a factor of two, a very curious factor of two. Now, of course, the experiment hadn't been done, so no one knew whether it was off by any factor. interestingly einstein did the calculation in 1911 with a very preliminary version of his theory of general relativity and he was motivated roughly by an application a qualitative application of the principle of equivalence and he ended up making the same prediction saldner made in 1911 And very fortunately for Einstein, no one did an observation of this effect.

1:40:00In the years between 1911 and 1915, when Einstein completed his theory of general relativity and found the full dynamical equations, he then redid the calculation on his full theory and got an extra factor of two that the deflection should be twice as strong. And he did that calculation in 1915, 1916. And then when the experiment was finally successfully done, the Eddington expedition in 1919, they found this factor of two. There was some historical scholarship that doubted whether the precision of the experiment was strong enough to make a discovery claim. I've talked to some historians in the last few years, and there seems to be consensus from at least some of them that they just had really good atmospheric conditions and the results on retrospect were robust.

1:40:49But in any event, that's what rocketed Einstein to international started. And that's why now you draw a circle and a wavy line on top and a wavy line below and everyone's like Einstein. That's what he became this huge character. But there's a really interesting point about this factor of two. So you can get the factor of two through a very detailed calculation. I assign it. So I teach general relativity at the graduate graduate course in general relativity at Harvard for like over a decade. And and I this is a calculation that's a homework problem. Now, it's it's it's amazing how in the history of science you go from things that that were huge, you know, you know, apocal discoveries to now just homework problem three.

1:41:30But there's a really interesting connection of this factor of two to an old question. The question about the equivalence of inertial mass and gravitational mass. I know this is a little bit of field, and I'm sorry, but, you know, Robinson, I've listened to your podcasts, and sometimes the most fun parts of the podcast are the tangents, the diversions. I hope everybody is on board. I've been loving the tangents those parts. So the thing that when a student takes a course in Newtonian mechanics and they learn about Newton's law of gravity, the student is told, oh, you know, that parameter, that constant proportionality, however, when I think of it metaphysically, the inertial mass.

1:42:07I'm sorry, another digression. Inertial, inertia. So, Robinson, last time we spoke, you asked me for etymology. Inertia is such a lovely word. It comes from, it's the same, it's the same as inert, which means, which comes from in, not, and ours, which is Latin for craft or make a creative crafting making. So inert is to not be doing anything creative and, and inertia is the quality of not doing that. Anyway, so, but when you take it, when a student takes a course in introductory Newtonian mechanics, they learn that the same parameter m, mass, m, inertial mass, that shows up in f equals ma, that relates net external force to acceleration experience by an object, is like the gravitational charge.

1:42:49The quality of an object that makes it both actively produce gravitational effects and passively respond to gravitational effects. and and that's quite surprising and it leads to various predictions like that objects that are small enough that they won't tug on the earth too much and the earth we can treat is still objects will will hit the ground at at the same time if they're dropped from the same height because the mass is canceled the inertial mass and f equals ma cancels the mass in the law of gravity just perfectly. And to Newton, this was a great mystery. I guess technically, it's not that they're exactly the same, but they're the same up to some universal constant proportion that we can absorb into the redefinition of Newton.

1:43:35Yes. I'm unaware of Newton saying it was a great mystery. I would have thought he would just say, there's a single thing, mass it showed up it shows up in one law in the fundamental dynamical law and it shows up in the law of gravitation um there's no mystery why something is itself right i mean if you think there you know there were like formally as it were different roles it's playing but you might not think it's a mystery you might just think well the same thing you know i've got this physical magnitude floating around and it it does a couple things right thank you tim you're completely right that was unjustified i'm not an historian i don't know for a fact that newton commented in that way on it is interesting to note though that of course later we discovered other kinds of charge like electric charge that show up in a very similar way in their appropriate force law the coulomb law electricity has electric charge that also you know objects have and it produces forces but electric charges is not like inertial mass.

1:44:36So it's weird that gravity has this feature. And so what a student is told, the more proper thing is, you know, you go to a class and the teacher says, you know, isn't it a weird coincidence these happen to be essentially the same thing? But that can't be the case in general relativity. It can't be the case that they're literally the same thing because of course in general relativity, there are massless particles. There is light and And light has zero, that parameter m, the inertial mass is zero. And yet light responds to gravitational fields. And light carries a kind of energy and energy gravitates.

1:45:15So there's clearly something going on here. When you take a course in general relativity or read any of the textbooks, this is basically not really discussed. The equivalence of inertial and gravitational mass in the Newtonian regime is mentioned as one of the motivations for eventually getting to the strong equivalence principles, the weak, strong equivalence principle. But then it's like not mentioned. And then you just have general relativity. We just use general relativity. We look at geodesic motion and we don't really talk about this anymore. But it is a really interesting question. Like what happens to gravitational mass and inertial mass?

1:45:49What happens to them in the full theory? And I don't know, Tim, have you probed this question? Because it's a really interesting question. I mean, I'll say a few comments. So look, we're on fleas on fleas at this point, but okay, let's just do one flea down. First, a comment for people who don't know, not only was there predictions of gravitational lensing, as it were, for light, if you'd had a corpuscular theory, but famously, I can't remember who, calculated the mass of a Newtonian black hole. That is a star so heavy that by its gravitational mass, light could not escape it. The escape velocity was infinite.

1:46:30OK, so, you know, you just couldn't get light off the surface of this star. Just for a second, it's John Mitchell. OK, Mitchell. And then Laplace popularized it. But yes, right. So, you know, yeah, I mean, you can ask these questions. it it leads people to have a very mistaken notion of what a black hole is because there you really do have a very massive object that's sitting there that you could you know fly into and crash into the surface of in a black hole there ain't no such thing right there's nothing there sitting in the middle that you can bang into um um so it's a very you know but i mean from my point of view, I guess, once you do the trick of thinking, gee, all these gravitational effects that Newton is explaining via F equals ma and the universal force of gravity, those effects are now going to be explained by the geometry of space-time itself.

1:47:26Then it's trivial that light and everything else that's in space-time, right, all you need is an object that's in space-time and where its dynamics is written down in terms of space-time, it'll be affected by gravity because gravity is space-time geometry. So one part of that kind of goes away, right? The other part is the source part, right? What about a light goes into the stress-energy tensor, which is important in solving the Einstein field equations? And for that, we also have to remember this famous quote of Einstein when he says that the field equation is like a mansion, one half of which the right side, the geometry side is marble, high grade marble.

1:48:15And there's wood. The other side, the stress energy tensor side is low grade construction plywood. He himself knew that that representation of the matter energy content was just a clutch. Yeah. Right. A convenient clutch, the one that had the right mathematical form to make the equation work. And he knew it wasn't right. This is also, you know, he talks about this very nicely. He knew that wasn't right. So when you get to that side of it, but how does the mass energy of a photon influence the gravity, which means influence the structure of space time? Because what we have is going through the stress energy tensor, already you're in a mess, right?

1:48:56So I don't think there's a general relativistic account of that because that's you're getting to the part of general relativity so we've never really complete right yeah so but there's some things you can do here so certainly inertial mass plays a role in relativity uh this is again getting a little in the weeds but but we do assign uh particles in general relativity inertial mass the the invariant you know momentum for momentum squared is is it determines it's a casimir determines uh the inertial mass of the particle. And then, of course, we have gravitating mass. I mean, after all, in the Schwarzschild geometry, the black hole geometry, but more generally the geometry around any kind of spherical object, there's a big capital M that plays a certain role in determining the shape of the space-time in the vacuum exterior of a spherical body.

1:49:45And the question is, is there some relationship between these apparently qualitatively different things? Now I'm puzzled. The Schwarzschild solution is a vacuum solution. There ain't no mass or energy anywhere in that solution. It characterizes something about the geometry. Yes, that's right. But if you, for example, so if you take like a star, not a black hole, like a physical star. Yeah, yeah, yeah. And you want to ask what is the space-time geometry and the vacuum exterior to the star. This is not internal, the exterior to the star. the schwarzschild geometry has a parameter in it right in the in the it's 2gm over rc that capital m plays a role in in in the theory uh if you tune it to zero then the then the geometry reduces to the minkowski geometry um so so here's here's just like a a a question you can ask right if there is some notion of gravitational sources and there's a notion of inertial mass how are they related on general relativity.

1:50:48And the first person who appears to have, at least as best I can tell from the historical record, precisified this question was Steven Weinberg. And he did it in 1964, which was an incredible year for physics. I mean, Tim, 1964 is Bell's theorem. It's, well, I mean, this argument, let me give in a moment for Weinberg. It's Penzias and Wilson's discovery of the cosmic micro background radiation. Murray Gell-Mann and George Weig proposed the quark model in 1964. Jim Cronin and Val Fitch discover CP violation and therefore time reversal violation in 1964, right? And the Higgs mechanism is proposed by multiple people, right?

1:51:33Higgs and all in 1964. I mean, what are we doing? Yeah, right. What are we doing? We're stuck. We are really stuck when you compare that to what's been going on recently. Right. But so Weinberg is studying the relationship between quantum field theory, which is still relatively young in the 1960s, and gauge theories and general relativity. He's trying to understand the relationships between them. And he's trying to arrive at something like the... Well, he's not intending to arrive at it, but it comes out of the calculation. He's trying to understand the relationship between gauge invariance and soft gravity.

1:52:12This is a whole complicated story that he was working on. But he ultimately arrives at this interesting statement that the gravitational charge, the thing that sources and feels gravity at the level of individual gravitons bouncing around in these diagrams is not little m alone. It's this particular combination. It's written in the 1964 paper. The formula, I'm going to say it, but like, but OK, it's not that complicated. It's twice the energy over c squared. you might have thought it was just energy over c squared because e equals mc squared it's not, it's twice the energy over c squared minus minus m squared c squared over e the energy it's these two terms, it's a difference of 2e over c squared minus m squared c squared over e and it looks, what a funny looking formula if you actually simplify it out for a particle that has non-zero mass, what you find is that it's equal to this relativistic Lorentz dilation factor, gamma times MC squared, like you would have expected, but there's a corrective, there's an additive corrective piece.

1:53:24It's actually that contribution times one plus V squared over C squared. And for very, very low velocities, the V squared over C squared goes away and the gamma goes to one and you get the usual Newtonian mass. But for a massless, even a particle that's ultra-relativistic, and there should be a continuity between a very light mass ultra relativistic particle in a particle that's literally exactly massless level of general relativity if you take the ultra relativistic limit the energy is so dominant it wipes out the second term and you get two e over c squared which means that the gravitational mass is twice what you would have thought the inertial mass was it's an extra factor of two and so the gravitational force is twice as strong on that ultra relativistic particles you would have expected exactly the factor of two you'd need for that correction.

1:54:10And although Weinberg did this with this quantum field theory calculation, you can do it yourself. If you go to the Einstein field equation and trace reverse it, and then plug in for the stress energy tensor, the stress energy tensor for a point-like particle, it's a delta function stress energy tensor, you'll just get this answer. It pops out. It's very beautiful. And it's closely connected. The M that it shows up in the Schwarzschild solution is closely connected. It's called the Comer mass. Anyway, so I just wanted to mention, it's a beautiful story and a little bit of a tangent, but one that I think isn't told.

1:54:37often enough. So now let me get to your points. So you're completely correct. The wave function does other things. It seems to be connected with a clearly observable phenomenon, this interference phenomenon. So I'm going to tell you another thing that happened. So I was playing around with these non-Markovian stochastic processes that I had just sort of stumbled into. And I noticed something a little funny about them. If you, so they had a very simple translation into a Hilbert space picture. And although I'm not going to be able to show the details and figures, Robinson, if you want, I can send a link to a talk I gave, a more technical talk where I show one way that one sort of naturally arrives at this.

1:55:23But there's a way to construct a very elegant Hilbert space picture that describes this process. And then something really curious happens. The whatever initial probabilities you might have been using, the initial epistemic uncertainty, the initial probability distribution for the initial configuration of the system that then you're evolving forward with this tocast a process, that gets represented as a square positive semidefinite matrix in this formalism. this is the density matrix and as you evolve it forward with this sort of non-markovian process this matrix goes from being diagonal to having off-diagonal entries these are the so-called in quantum mechanics will become coherences and usually we talk about coherences in quantum mechanics they're just primitive things we don't there's no interpretation directly for what the weird off-diagonal entries of this funny looking matrix are But if you take one of these processes and you impose an event, a Markov event, a moment in time when it becomes Markov and forgets its past, the entries in these weird off-diagonal coherences, they vanish.

1:56:41um and this suggests a reducible meaning for them that they that they're describing as a kind of history or non-markovianity in the process and once you see this happen it makes you it makes you a little bit you're like oh wait a second usually we say that there's no meaning for these things at least on this kind of a model these weird off diagonal entries these coherences seem to have some other meaning that we can give them. This is all very suggestive, to be clear. We're not yet talking about interference.

1:57:17But, and now what I'd like to do is actually give the simplest explicit model I can that would motivate this indivisible stochastic picture. Before you do that, let me just try and get a little clearer about what you just said. Yes. And I'm hearing this for the first time. Yes, I know. It's all very new. Right. So we're all in the same boat. I mean, what I think I heard you say was that you have this. OK, you have the stochastic process. Fine. And instead of a precise initial state, you used a probability measure which represented some epistemic uncertainty about. Sure. But that you could you could set it to one.

1:57:59It's not necessary. Yeah. If that's true, then I would expect all kinds of things to go away because that's just conditioning. That's just standard Asian conditioning, right? I make a measurement and all this epistemic, a bunch of epistemic certainty goes away instantly because I found out among all these possibilities, at least cut down on them. So when you say all of these terms, you know, if I have this later event, all these terms disappear. This sounds like what I would have expected if they were arising from an epistemic source. I can make this more precise after I give a little bit of an example.

1:58:37So now I'm going to give this example I've been referring to, a very simple example that will illustrate some of these ideas, and also kind of in a sense how cheap Hilbert space pictures and quantum-looking descriptions can be in some situations. So imagine the simplest dynamical process that we would call a dynamical process. Deterministic, time is discrete, time only happens in time steps. Let's say it happens every microsecond, something changes. And this system has a finite number of distinguishable configurations. Let's say 17, because we used 17 earlier. let's say 17 a nice big clearly not special number everyone knows the smallest uninteresting number is 26 by the way but okay so 17 um so we've got some system you could think of it in many ways maybe it's a strange i guess 17 is not the number of sides in the platonic solid but maybe someone has some weird uh non-platonic solid die that has 17 whatever you want to model um and i guess a die is a bad example because that would look too probabilistic this is something that is just changing in a deterministic way.

1:59:50It starts in some configuration, let's say three, and then it goes to seven, and then it goes to four, and then it goes to 13, and then it goes to one. Probably your automaton is what you've got, sort of. Yeah, yeah. It's a very, very simple cellular automata. That's one way to think of it. Every cellular automata is a dynamical system of this kind. Just cellular automata, usually we think of it as like a linear array or a two-dimensional array or more of cells. I want to keep it even, I don't want to use any of that language. It's super simple. Okay, so this system is just going through these things.

2:00:16And let's assume the dynamics is so simple. It's logically reversible, which just means that from whatever configuration the system is in, the law will tell us what it will be in next and will also tell us what it will be in before. so we get a nice one-to-one mapping, you know, and we can predict exactly what the system's going to do. For this kind of a system, its behavior will end up having to be cyclic. Either it hits all 17 configurations and resets, or it doesn't hit all of them. If it doesn't hit all of them, just throw out the extra ones. But just assume it hits all of them, okay? So it goes through some cycle and it returns to where it started.

2:00:54Couldn't be a simpler process. And every step happens at one microsecond. So here's the thing we can do with this process. And this goes back to a question I had a very, very long time ago. Because, you know, when you're learning computer programming or learning elementary physics, you think about discrete time deterministic processes and you wonder, you know, can we interpolate them? Is there some smooth way to interpolate this to continuous time? Wouldn't that be interesting if there was some very natural way to do it? It turns out there's a very natural way to do it. As best I can tell, I think I'm the first person to write this down, although I'm not sure.

2:01:29I don't want to claim priority because it's very hard to search in the literature for whether an abstract idea has been done if it hasn't as of a name. So probably it's been done before. But here's the thing you can notice. We can represent each of the 17 configurations of the system with a vector, a vector from what you'd call the standard basis. So imagine you've just got like, you people are probably familiar with 3D space. You've got the three basis vectors, I hat, J hat, K hat, right? it's a 17 dimensional space now with 17 orthogonal perpendicular directions each of them has a basis vector pointing in that direction they're all perpendicular to each other now it goes without saying that this is not a real vector space like it's not i mean physically real it's just it's just it's a mental construct that we're just going to use and now we can think of the system's configuration changing as a transition from one vector to another right it goes from configuration from three to seven, that means it transitions from the third basis vector to the seventh basis vector.

2:02:31What's really nice about this representation is that we can write down the dynamical law as matrix multiplication. We can introduce a matrix. It's a 17 by 17 matrix. It's called a permutation matrix. You can check that it's just made of zeros, except it's got a bunch of ones and every row in every column is exactly one, one in it. And if you look at this matrix and look at all where the ones are, you can figure out what the rules are going to be for the system. The locations of the ones in this matrix tell you exactly what the dynamical law is. And then, of course, each of our basis vectors will be written as a column matrix with zeros, but then a single one in whatever.

2:03:07So the first basic vector is going to one in the first entry. The second is going to one in the second entry, the third and the third entry. Otherwise, all the entries are zero. And then we can model this dynamical process mathematically as just repeated multiplication by this permutation matrix. You just multiply it again and again and again. If you want to do one step, you multiply once. If you want to do two steps, multiply twice. And in general, if you want to do n steps, which is n microseconds, we're assuming that each time step is one microsecond, you just multiply your starting basis vector by the nth power of this permutation matrix.

2:03:40And this is a very beautiful story. And there's some beautiful mathematics and talk about the eigenvalues of this thing, the roots of unity. There's this whole very beautiful story. But now I want to do something a little bit different. this permutation matrix as a permutation matrix a matrix that's made of zeros and ones where every column and every row has exactly one one otherwise all zeros is in particular a unitary matrix so the word unitary there's no avoiding mathematics here i'm sorry the math will be necessary a unitary matrix is a square matrix with the property that if you flip over the matrix along its main diagonal it's top from whose point of view am I talking about?

2:04:22If you're looking at a matrix and you look at the upper left corner and you draw the diagonal down to the lower right corner, you just flip the matrix over. And if the matrix has any complex numbers in it, which this certainly does not, but if it did, you'd have to complex conjugate also. And if when you do the double step of flipping it or transposing it and complex conjugate it, if what you get is the multiplicative inverse of the matrix you started with, we call the matrix unitary. Unitary matrices play a very central role. Earlier when I was talking about the Dirac-Vinom and Axioms, I said that the evolution when a system is not being measured is unitary.

2:04:52That is to mean that it's described by a time-indexed family of unitary operators or unitary matrices of exactly this kind. And of course, this is not an accident. Notice we have a time evolution being carried out by a unitary matrix here, although in discrete time steps. And this permutation matrix is a very trivially unitary matrix. It satisfies the right conditions, but not in any fairly simple way. Here's the amazing thing about unitary matrices. We can take them to any power we want. And a result is still a unitary matrix. So I did talk about taking this permutation matrix and swearing it or cubing it or taking it to the nth power.

2:05:34Integer powers, these correspond to integer steps, but I could also set it to some fractional power. I can set it to some power that's between 2 and 3 or between 7 and 8. In fact, I can set it to any smooth real number power between those integers. And when I do that, the result is still unitary. It's not still a permutation matrix, though. In general, it will have complex numbers in it. Maybe people who are listening know that when you take square roots of things, you can get imagined numbers sometimes, complex numbers, things like that. Famously, the most elementary imaginary unit, i, is the square root of minus one in a formal sense.

2:06:11So you get now at any time, smoothly interpolated time, any time you want between the integer microseconds, you have a unitary matrix. And if I evolve, if I multiply my original basis vector by this unitary matrix at one of these intermediate interpolated times, I will in general not get a basis vector. I will get a linear combination or a blend or a superposition of the basis vectors with, in general, complex value entries. And at this point, your thought might just be, well, that's curious. Whatever. Put that aside for a moment. We could do something else with this unitary but no longer necessarily real matrix, the 17 by 17 matrix, the permutation matrix, but set to a power that's some not integer power, some unitary matrix that complex numbers inside of it, we can do something else to it.

2:07:18We can define a new 17 by 17 matrix by modulus squaring every individual entry of the unitary matrix one at a time. You modulate the square, the upper left corner entry, and stick that in a new matrix. You mod square the next entry. You put that in the next entry of matrix. You can define entry by entry, entry-wise, a new matrix whose entries are all mod squares of the individual entries of the first matrix. The matrix we get when we're done has a very interesting history. Matrices of that kind were first characterized in the math literature, pure math literature. Real Analysis by Alfred Horn in 1954.

2:08:02He called matrices like this, a matrix whose entries are all non-negative real numbers, but whose entries are entry by entry, the modulus squares of a unitary. He called those matrices orthostochastic. In 1989, they were renamed by Thompson in a lecture that I can't find. By the way, if anyone is listening and can find the text of this lecture, I've been looking everywhere for it. In 1989, Thompson gave a lecture. uh johns hopkins it was a johns hopkins but i can't find the text of the lecture anywhere in which he renamed these things unistochastic and now orthostochastic is used to refer to the case in which we don't start with a complex unitary we start with a real unitary or orthogonal matrix so orthostochastic matrices are are a special case where the matrix we mod squared we didn't even it It was already real.

2:08:55But if we start with a complex matrix, then we call the matrix we end up with, this 17 by 17 matrix with non-negative entries we've constructed. It's called unistochastic. And you can show that the matrix we get is a stochastic matrix. I said unistochastic, and the reason for that is a portmanteau of unitary and stochastic.

2:09:16By the way, portmanteau is a portmanteau. Isn't that funny? From Takari and Mantle. It's okay. It's funny. But we can write that in a matrix. We can check that this matrix is, in fact, a stochastic matrix. Now, if it is stochastic matrix, that means that you have a matrix whose entries are all non-negative real numbers. And at least in this convention, we're going to look at columns. Each column sums to one. In the stochastic process literature, sometimes people work with what are called rows stochastic matrices with a row sum to one. And it's just a matter of convention. But the one we get out of this is it's column stochastic, but actually it doesn't matter because it turns out that for a unit stochastic matrix, it's also row stochastic.

2:09:58It's called doubly stochastic or bistochastic, but in particular, the columns are stochastic. The columns sum to one, I'm sorry. And what's fascinating about this is that a stochastic matrix defines a stochastic process. what i was doing before where i took my basis sector and multiplied by this weird unitary i'd found at intermediate times i could instead have multiplied my starting basis vector with this unistochastic this real non-negative stochastic matrix i've just constructed that just magically comes out and if i do that it also interpolates between the microseconds. But it doesn't interpolate with complex numbers or mysterious complex superpositions.

2:10:52It interpolates it with just a probability distribution at each intermediate time. So the picture that emerges from this interpolation of my discrete time process, it was discrete, it was deterministic, is I've now interpolated it to continuous time where between the discrete time steps, we have a smoothly evolving probability distribution. And at each discrete time step, the non-trivial probabilities, well, it starts off with no probability, the probabilities are all trivial. And then as we go between the time steps, we develop these non-trivial probabilities, this non-trivial probability distribution.

2:11:29And then when we get to the next time step, the probabilities all snap back into being deterministic and we just have a one and then zeros again. And so we have this very funny evolution where between the discrete time steps, we have an interpolation by a stochastic process with non-trivial probabilities. And at every single discrete number of integer numbers of microseconds, the system returns to being in a definite configuration. And you might ask, what kind of stochastic process is this? Is it a Markov process? The answer is it's not. It's not a Markov process. It's a lot like the kind of process that Shoemaker's universe had.

2:12:09It's a process where we can't divide up the evolution. The evolution just fails to divide between the integer time steps. So we have this evolution that is sort of indivisible between the time steps, and then it divides at each time step. It is an indivisible stochastic process. It's the simplest way to see these things emerge. I didn't give them, I mean, I did name them indivisible, but it turned out that the name had been given to them by Simon Mills and Kevin Modia in 2020, a couple of years before I arrived at them. So some interesting things to say about this. One is we get this very beautiful interpolation of our original process.

2:12:48We can now talk probabilistically about what's going on between the steps. We see that at these division steps, every integer number of nanoseconds, the weird complex superpositions that we would have seen if we were evolving forward with unitary, they disappear. They disappear at these special moments. What's also really interesting is, and now this is to come back to the complex version, I could again have evolved it forward with the complex unitary matrix, and then I would have had to introduce a Born rule to modulus square the entries of the state vector to get the probabilities. That's the other way you could have described the evolution, if I describe it that way, using this complex unitary where I have to do this mod swearing at every step to get the probabilities, if I do it that way, then the evolution is formally Markov.

2:13:43The unitary neatly divides. You just can multiply by any duration of time you want. If you give me the complex vector at any intervening moment, continuous intervening moment, You give that to me, I can evolve from there with no, I don't have to do anything about the past, using this complex unitary. By introducing this alternative unitary complex description with these exotic features, superposition and so forth, we get what looks like a quasi-Markovian description. And we get more. I mean, there is, after all, a unitary matrix, smoothly interpolating time that's carrying this vector forward in time.

2:14:22And you can write down its generator. You'll find it's generated by a self-adjoint matrix, the Hamiltonian. That matrix has eigenvalues, energy eigenvalues. And so we actually get many of the features we would have expected to have seen in a quantum system. A Schrodinger equation, the system has a differential Schrodinger equation now. But none of these ingredients have to be interpreted as physical ontology. It emerged out of what is the simplest discrete deterministic model with the most straightforward interpolation. interpolation. We can either view that interpolated model as an indivisible stochastic process in which the process is not Markovian, but there's also no exotica.

2:15:09There's no complex numbers. There's no wave function. There's no superposition. There's no Schrodinger equation, none of that stuff. And we'll get all the right predictions. Or we can introduce this alternative Hilbert space picture, which is mathematically equivalent. There's a correspondence. This is the stochastic quantum correspondence. And in this Hilbert space picture, we have a much more convenient evolution law where we can evolve from any time to any other time. The unitary matrix that carries at the evolution is way easier to use. We have a Schrodinger equation, we have eigenvalues, we have all this lovely stuff.

2:15:46And it's on this model that if you began with a non-trivial epistemic probability distribution for your initial state, then rather than a vector evolving, thing, you would use a density matrix. And the density matrix ends up being diagonal at these special division moments, and then it becomes not diagonal. It develops these weird coherences in between them. If you understand how this simple model works, and I'm describing it waiting my hands around, it is much less work to just write down. But if you understand how this works, it's very suggestive. It gives a strong hint about what could be going on in quantum mechanics.

2:16:22This is a model. I mean, it's a discrete model, but if you increase the dimension from 17 to a Googleplexplex or, I don't know, Ackerman's number or some huge number, you begin to think, well, could you model any quantum system this way, really, to any level of precision? Now, what I haven't said, I haven't talked about interference yet. I haven't talked about the measurement process or non-commutative observables. This model has beables, the ways that it can be. Those are labeled by the 17 possible configurations. But in quantum mechanics, we have theories like the Bohm theory, other kinds of observables we can measure.

2:17:06These are emergent interactions from the process of a measuring device interacting with the system. I call them emergibles because otherwise you have to call them non-beable observables and I think there should just be a name for them. And so what we can talk about in a moment, if you'd like, is how all those other observables can show up, how the measurement process shows up, how interference shows up. But before I go any further, I've been talking for a while, so I'm going to stop. But I just wanted to stop now that I've written that model and given you a sense of why you might be compelled to think that this could be so interesting.

2:17:39Okay. So can I respond for a little while? Yes, please. I'm sorry. Yes. This was again, this is the first time I've heard this. I followed a bunch of it, I think. So let me explain in a different way what you at least described at the beginning and why a bunch of stuff that you said didn't surprise me. I saw it coming and I'm not quite sure how we've advanced the situation. So let me just you started with 17. But for simplicity state, let's take a three state system, right? A nice little cellular automaton. It only has three possible states and time is discrete. And at every time moment, it moves from one state to another in a deterministic way.

2:18:26It's going to go in a loop. Just a small termological thing. State is fine. I usually prefer configuration just because the word state can sound like quantum state, and I just want to be a little bit. That's fine. But you can say state, just for people listening. I mean, if you're thinking in terms of automatats and machines. Yeah, we do call them states. That's fine. Yeah. You know. Yeah. If you think, if you know Turing machines and stuff, you can just think of these as machine states. And the program is telling you that A goes to B and B goes to C and C goes back to A. This thing was supposed to be deterministic in both directions, so it's got to go in a loop.

2:18:58It's going to be a three-step loop. You could have one that's stuck and the other two bouncing back and forth, but let's do a loop. Okay, good. And now I specify the entire dynamics just in terms of that state transition, which you could do in a matrix, the very matrix you described, a bunch of ones and zeros that say this state goes to that state, this state goes to that state. All of that I can use a matrix for. It'll be a very special, quite simple matrix. Um, and, and I can visualize this in, in a, in an abstract three-dimensional space, which is what you did and say, well, let me take a space and let, let, you know, I've got three orthogonal vectors, but this one represents state a, this one B, this one C, the dynamics tells me, sorry, this one goes to this one.

2:19:47This one goes to this one. This one goes to this one. Okay. So I could do it that way. That's for discrete time, right? At each time step. Again, it's very much like a Turing machine. At every time step, it does a straight transition. Now you say, well, gee, what if I wanted a kind of intermediate states? Well, I've got these vectors. I could say, well, let's just let this vector kind of smoothly at an even space, you know, pace, rotate from here to here. This one rotate from here to here. This one rotate from here to here over a microsecond. Now I have a continuous time evolution. It's going to give me the same thing at all the, you know, microsecond things.

2:20:30But now I can say, gee, at half a microsecond, this guy's halfway here and this guy's halfway here and this guy's halfway here. Those don't really represent actual possible machine states, right? They're kind of superpositions, you could say. Formal superpositions at that level, yes. Right. That's going to be that's going to be now you then have another thing that happens, which is you say, well. This is on the assumption that actually at every microsecond, it's in exactly one of these three states. What if I'm unsure? What if I think there's a 50 percent chance it's here and a 25 percent chance.

2:21:05Now I can stick a vector. I'm now mixing epistemic with other kinds of probabilities. And you have to be very careful about the term probability here. and that epistemic thing will be driven by this underlying dynamics and it'll move around right and i guess i can imagine although i couldn't see this right off the top of my head that if i look at at how these kind of states that are not in the definite directions go it might not be markov that might not be markov i'm not sure right yeah in general it's not markov that's right yeah so that might not be markov right right that that's okay yep um So I got all that.

2:21:46Yep. I think. Yeah. Yeah. That's basically. The worry was interference effects. I don't. We didn't get there. Because there ain't going to be no interference. I didn't get interference yet. You know, that's why I'm puzzled. I mean, there was a lot of math that went into that. And I think I was following most of it. And I seem like it's kind of easy. I mean, I think what I just described can give people a picture in their heads. Sure. We now have this vector moving around in a continuous three-dimensional space, whereas what's really going on is only a three-point physical space, right? A three-point state space, and you're just moving at times between them, and then I'm putting all this other stuff in.

2:22:26Yeah, I can see how you can do that, and I can see it would have a lot of the properties you gave, but I'm not quite seeing where I'm getting out of that. Okay, so I'm so glad you have a picture. I didn't pick three, by the way, just because I didn't want to confuse anybody watching that this is like physical 3D space. Right, yeah, yeah. Yeah. I picked three, so I can only do it in three. That's the thing. Exactly. Right. If you have tesseract hands, you should get that checked out. Or 100 billion or whatever. Right. Okay. So, no, your picture is right. On the one hand, so let me just quickly address your question about probability here.

2:23:05um the the simplest way to talk about the probabilities that are so there's there's two ways to think about the interpolation and you you illustrated both of them one way to to to think of the interpolation is a probabilistic interpolation where the vectors are kind of if you want to visualize them in this abstract space they're kind of rotating into each other but not really a rotation they're also deforming or stretching a little bit because because the coefficients have to add up to make one. They're probabilities. They're all non-negative. That's one way to think about it at the level of what the individual vectors are doing.

2:23:40And this is, when you write it down in mathematical terms, it's exactly the form of one of these indivisible stochastic processes that we characterize in literature. They're not Markov. Or we could think of them as rotating, really rotating, like really rotating, but then the coefficients aren't going to add up to one. You have to square them and then they'll add up to one. And that alternative interpolation is Markov. It's like rotation. You can divide it up. It's nice and Markovian looking. That's a lovely way to think about it. In general, when you do this, though, the entries will fail to be real valued.

2:24:19They'll entail complex numbers. So it's not just rotating in ordinary 3D space with ordinary numbers. You'll need complex numbers as well. But of course, what you can do if you take this complex now vector sticking in some intermediate direction and you mod square its entries, you get the vector from the first version. And so the flat-footed way of saying is, okay, what's going on here is I have interpolated my deterministic process at discrete times to a stochastic probabilistic process interpolating between those discrete times. And I can represent it either with a unistochastic matrix, then it looks like an indivisible process, or I can use complex numbers just as a useful mathematical tool.

2:25:00And then I get all this other beautiful, wonderful stuff. But I also have a lot of exotic new features like complex numbers and Schrodinger equation and unitaries and all that stuff. So your picture sounds like you get it completely. This is great. Okay. Can I just make another? I was going to say this at the very beginning and I forgot to say it. But just so people understand, when we're talking about the mathematics, there's a thing called a probability measure. That's just its name. It's a measure. It has to be normalized to unity. To be clear, measure is not measurement. It's a mathematics.

2:25:39If I have I have a space, if I have two pieces that don't overlap, the measure of their union has to be the sum of their measures. OK, so there's some formal features that a measure has to have to be called a probability measure. But the word probability could really mislead you there. You know, if you're just talking about plain old statistics, they have to be described by a probability measure. They have to add up to one. Right. Percentages have to add up to one. They can't be negative. You know, the percentage that are in the union of these two groups has to be the sum of the percentage in each group.

2:26:17So you use them to describe statistics. You also use it to describe a stochastic process, meaning a non-deterministic process is a dynamical notion. Yes. You also use them. Many people use them to describe people's beliefs. They use probability measures to measure credences. I myself think this is a bad idea, but pretty much everybody does it. Okay. And they're all mathematically probability measures, but the use of them doesn't mean that you're introducing stochasticity or denying determinism. Agreed. You really have to track when you say that's a stochastic process. I mean, the process we started with was a deterministic process.

2:27:03And even with the interpolation. It really wouldn't normally be what I would think of as stochastic process. Right. Where from a given state, there are probabilities, there are definite probabilities of how it will evolve. Right. It's just give me this state. Okay, it doesn't have to go this way, but there's a probability in the sense of a number between zero and one or a probability density that it do this, that it do that, that it do that, right? And so even though we talk about, and at the level of matrices, you won't see any difference because this is all the same math. It's all just probability measures.

2:27:43So one does need to track what the hypothesis is here. One you really want to track is are we giving up transtemporal determinism or not? Can some complete initial state evolve differently or is there only one way it can evolve? How constrained is it by the laws? I have to just say, this is one of the reasons why I'm enjoying talking with you so much, because your insistence on precision. So when I think about analytic philosophy, I'm not going to define the definition of analytic philosophy. But one of the things I think is really characteristic of analytic philosophy, when I'm talking to scientists, to physicists, and trying to explain to them why it's worth their while to learn something about how to think a little bit like an analytic philosopher, is that eventually the experimental data and the mathematical symbols dry up.

2:28:38If you're a pure mathematician, that doesn't necessarily have to happen. But if you're doing science or physics, there's always going to come a moment when the mathematical symbols and the experimental data dry up. You can't just say, I don't need any, just go with the math and the experimental data because at some point it stops. And we're seeing this exactly right here. I gave you a bunch of mathematical representations, and you did exactly what an analytic philosopher is great at doing, which is, okay, well, then what? When the symbols dry up, how do you get to the next step? and good analytic philosophy is learning how to understand what it means to try to be rigorous after this mathematical symbols and the experimental data dry up it's one of the lessons i try to convey when i talk to to physicists i i teach philosophy and physics classes and i teach my physics classes i'm especially trying to get across to the physics students why it's so important not to think that you give up on rigor the moment this mathematical symbols dry up.

2:29:38So I'm really glad that you brought this up. Can I just make a little comment about this? This is a really important case for quantum mechanics because a whole bunch of physicists were under the mistaken impression that when things decohere, which you can kind of define mathematically, you're entitled or allowed to suddenly take something that has nothing probabilistic about it, really probabilistic about it, and interpret it as if it's just epistemic uncertainty, right? And allied the measurement problem in that step, which is completely illegitimate. Completely illegitimate. I agree with you, yeah.

2:30:21Yeah, I'm not saying it to you. I'm just saying this is more than just nitpicky stuff. I mean, people being unclear about what the probability measures are that they're throwing around mathematically has been the source of a tremendous amount of confusion about the problems with quantum theory because they thought they could solve them by getting a certain number to look like another number and not seeing that. But to interpret it this way and then to interpret it that way, that's not automatic, even if the numbers look similar. I just wanted to point that out. When I was talking about the two different kinds of evolution in the Dirac-Vinomen axioms, and I said that they're categorically different.

2:31:00One is not a decoherence is on the Hilbert space side. And you can't just say, well, there's decoherence. The density matrix looks kind of like it would if we were imagining probabilities and therefore their probabilities. So everybody who's listening to this discussion should just stop right now, just pause and go find Tim's paper, Three Measurement Problems, which is one of the paradigmatic papers in Quantum Foundations. It's 1995, I think it's in Topoi, the journal, T-O-P-O-I. And you identify exactly this elision, this mistake, and you describe it as mistaking necessary versus sufficient conditions.

2:31:40And this is a thing that happens constantly in the scientific literature is people confuse necessary versus sufficient. it is the case that if somehow the entries of your density matrix have become probabilities in the right sense classical-ish probabilities then it is certainly necessary that it should look diagonal or whatever but it's not sufficient right and and this confusion of necessary versus you just you you identify it quite precisely and it's exactly it's when when i try to explain to physics students an example of the necessary versus sufficient condition mistake i often point to that exact point that you make in that paper.

2:32:18So anyway, so getting back to probability, I want to make very clear here, I'm not talking about a physical process yet. I'm not saying that nature has some discrete process in it and I've derived probability. I am 100 % not saying that. I'm only saying that if you're a student and you're like, huh, I could model a thing like this, could there be, not this process, but could there be some probabilistic process that agrees with my deterministic process at the nanosecond, the integer nanoseconds, but is a probabilistic process and has probabilities in between. What kind of process would that look like?

2:32:57Is there some very simple construction, right? And although it took me a little while to explain it because I'm using words, I feel like a nominalist. I'm like trying to explain everything without using math.

2:33:10But if you like write it out, it's actually like a few steps. It's incredibly simple. It gives a very nice analytic interpolation. And if you wanted to regard the true process, if you wished, as a probabilistic process, you could. And then it would be kind of up to you how you wanted to make sense of the probabilities. You're completely correct. It's not enough to say I've got real numbers that are non-negative and add one. When I talk to physics students or math students and I ask them, what's the probability? They'll say, oh, that's easy. It's defined by, I mean, it depends on how much math they've seen.

2:33:45But, you know, they used the 1933, which is curiously after quantum mechanics was formalized, 1933 Kolmogorov measure theoretic definition of probability spaces. They're just like, oh, to have a probability means that you have a probability space. You've got a measurable space, and you've got a distinguished collection of subsets. The sigma algebra satisfies various set theoretic requirements. And then there's this map on that sigma algebra to real numbers between zero and one. And it satisfies normalization and additivity and all these various things. And that's the rigorous definition. And I say that's not the rigorous definition.

2:34:22That's not the rigorous definition because that applies to if you take Earth and imagine slicing Earth up into chunks and you just treat every chunk fractionally. How much of the Earth is this chunk? That will also satisfy the same axioms. And it does make it a probability, right? The math has run out and now you need something after the math, right? You haven't given me a rigorous definition of what a probability is. So I 100 % agree with you. 100%, sorry for the pun. you could think that the process you're starting with is describing just statistical behavior of many systems you could think of it as describing an objectively chancy process another paper i recommend people read is again sorry tim but your your i think is your 2007 paper um what object what could be objective about probability i think it's the paper it's it's a lovely paper you talk about the different ways we could think about chancy probability or i guess you could think you're modeling a process where the probabilities are really standing in for something like credence, subjective credence.

2:35:22It's up to you. It doesn't really matter to me. For my purposes, I'd like to think of this in the objective chance sense, that the probabilities I'm using here, the stochastic process I'm describing, are talking about objectively chancy laws. But to be clear, I'm not taking a discrete process and then derive, I'm starting the way I'm saying that we could imagine there was some underlying process the whole time. um okay but now let me move on to i think we're are we good there or should i move on can i move on to the next step i mean i think i think everything seems like it's perfectly agreeable okay so you can start to ask some questions now i mean i have this probabilistic process i it could have as many configurations as i want like you do pick three i pick 17 it could be google plex whatever i could make the intervals between these markov moments longer than a nanosecond i could imagine a process where they're separated by a million years i mean whatever um and now i can ask some some questions uh what if my configuration each configuration so that there's a three configurations 17 configurations a googleplex configurations take a set of all of them we call that a configuration set or configuration space it's just the menu of the possible arrangements uh and then let's imagine that what we're describing really is a myriologically composite system of some kind so we're gonna think of each configuration as an ordered n tuple fancy language just an ordered list okay of the configuration of the 127 systems, something like that.

2:37:08Okay. So every configuration is really telling you a list of all the configurations of 127 different constituent systems. And we can get a discussion, are these elementary? For our purposes, they're effectively elementary. Last time Rob and I talked, we talked about the fundamentality of nature. I'm not going to, But we're just modeling. We could assume these are the muriological atoms of nature. We could assume they're coarse-grained. This is immaterial for the rest of purposes. Okay, so, and if each of them has, you know, some number of configurations, then the overall size of the configuration space is going to be some combinatorial, you know, how many configurations the first system has times how many configurations the system has, and so forth.

2:37:50That would be the overall number of configurations of this whole system. okay um and now what we can do is we can write down these we had this stochastic matrix it's made of these uh these conditional probabilities that we think of it as a stochastic process the conditional probabilities will not be marco they won't be divisible slightly different notions and now what we can do is we can start asking some questions about what happens when we marginalize them. So in classical probability, if you have the probability for some collection of objects, such and such is the probability that this collection of objects will have this overall arrangement.

2:38:35There is just an ordinary probability notion of ignoring all but one or a few of them. This ignoring is called marginalization. Formally, it's a sum, but I'm trying to avoid mathematical language, but it's just conventional probability stuff. And we can ask, okay, once I've done that, what happens? Do I get anything interesting? And what's really interesting is that with very, very simple models, in fact, I have one that I'm hoping will end up in a paper at some point. I just literally haven't had time. It was part of a lecture note that I gave. I can also send and Robinson send you a link to it.

2:39:16I do this for a very simple, very simple problem. Suppose that my system, forget all the complications, suppose I just have a two configuration system. This is ultimately going to model a qubit. Sorry about etymology, but do you know where the word qubit comes from? I assumed it was from quantum bit. Interestingly, that's only half the story. So I have this on good authority from one of the former students of Ben Schumacher at Kenyon College. Ben Schumacher and Bill Wooters work in quantum information, and they coined the term qubit 1991-1992 because they didn't have a name for a two-element quantum system.

2:40:00Bit itself is a portmanteau of binary information digit. John Tukey introduced that in 1947 and then was popularized by Claude Shannon, Shannon of information theory. And they were joking how Bill Wooters at Williams College and Ben Schumacher at Kenyon College, they were driving back to the airport in Ohio. uh ben schumacher i think was driving bill luter's back to the airport and they were joking wouldn't it be funny if we call these things cubits because in the bible there's a unit of measure to call the cubit oh and it's written c-u-b-i-t right and it's supposed to be the distance from you know your elbow to the tip of your index finger and if you read the bible when i read you know we read uh you know the the the story of of noah and i'm like it says cubits in it i'm like cubits what's going on here you know and because c is for classical you see because what's more classical than the bible right of and so they just replace the c with a q and and and david merman who probably people who think about quantum foundations and read about it probably know that he's a physicist who thinks a lot about quantum foundations has always thought this was a a uh a monstrosity of lexicography or whatever, you know, Q, U, and then a consonant.

2:41:15That's not a thing we do in English. He insists on writing capital Q and then B-I-T with no U in it. But the whole joke was that it was supposed to be a play on the biblical qubit. That's why it's Q-U-B-I-T. So just funny story. So imagine we're now going to do this story. I just told you that. Just told you about the story. And suppose that we're going to consider a system with just two possible configurations is going to model our qubit. And we let this qubit evolve according to this indivisible rule. And in the simple version, and Tim will be the first to say that the details of the double slit experiment, like the full fine grain details, are actually quite mysterious.

2:42:01And when people describe the two slit experiment in quantum mechanics, they never give you the detailed full story. They always simplify it in some way. And I'm going to simplify it because it's sufficient for my purposes here. So imagine that we've coarse grained our whole experiment so that we just have these two configurations. Think of them visually as upper part of chamber, lower part of chamber. Think of it as a super coarse grained version of the doubles that experiment. And this thing evolves from time zero. And then there's an intermediate time. This intermediate time we'll call T prime.

2:42:36and T prime, this intermediate time, is where the holes are if you want, but really we're just going to, you know, the system is either upper or lower. It's definitely one or the other because this is just a classical stochastic, I mean, it's ontological classical. At any moment, it's always one or the other. And think of this as it's either in the upper hole or the lower hole at T prime. And then at some final time, Tf, T final, we can ask about what the probabilities are. The model I gave you lets you compute probabilities whenever you want. We find something really interesting with this model, super interesting.

2:43:12We find that if you tried to write down, remember I gave you this underlying stochastic quantum whatever, you might naively ask, I mean, not Tim, you can ask this, but a person who looked at this would just say, well, wait a second. I have a unitary time evolution matrix that takes me from any one time to any other time. If I mod square its entries, I get a stochastic matrix. Why can't I just use the stochastic matrix I get from all the intermediate? Why can't I just take every intermediate unitary and just use the stochastic matrix from all of them and just multiply them together? Why can't I just do that?

2:43:51If you do that, you get the wrong answer. And you can see it very clearly in this example. If I naively compute the probabilities for the qubit or the two configuration system to be in either one hole or the other. I just compute those probabilities. And I pretend that from here, I can then evolve the system forward as if it were Markov, ignoring the past. Then I will get the quote unquote classically expected probability distribution at the end. But if I say that's illegitimate, the process is indivisible. I'm not allowed to just slice up my stochastic matrix at intermediate times. That wouldn't properly reflect what's really going on in the process.

2:44:36If I evolve the system correctly, not assuming there's a Markov event in the middle, I get a different distribution of probabilities at the end. And if you subtract from the correct final distribution, from the truly indivisible process, you subtract the Markov approximation where we pretended we could just, if you subtract the two, then the discrepancy is mathematically exactly the same as the formula for the interference effects. Like literally exactly the same formula. And when you look at descriptions of the double slit experiment, for example, the beginning of Feynman volume three, where he introduces quantum mechanics of double slit experiment, I don't remember if he says it explicitly or it's implicit in the discussion, but at some point he says, well, look, the thing either goes through one hole or the other.

2:45:26Suppose it goes to the first hole. Then what would the distribution look like? Okay. Now suppose it goes to the second hole. Then what would the distribution look like? And now we just average over the two. But that's explicitly making a Markov assumption. if you simply don't make the mark of assumption if you and you don't have to do any work here it's not like you have to to write down some very complicated bespoke thing if you just take the simple model i gave you i gave you that super simple model and you just let the evolution be just that unitary that we got without any any real effort you'll get the wavy interference effects without having to sit and say there's a wave there now the first time i saw this happen it was very surprising to me because like everybody, I'm like, well, I mean, if it sounds like hoofbeats, think horses.

2:46:12Of course, every time you do the experiment, you don't see the waves. This is important to make note about the double cell experiment. You do the experiment, one particle lands. You do the experiment, one particle lands, and over 10 ,000 run to the experiment, you get a distribution of dots that show certain bands where they tend to like to land and other bands where they don't. And from this, we infer that something wave-like has been going on. And like many people, I assumed, okay, that must mean that there's just something wave-like in this picture. it gets complicated once you have if you're going to send two or three particles into the experiment at once then it gets very complicated because with three particles the wave function lives in a uh nine-dimensional space and the holes are but but the point is when i first saw that a process that was just this simple discrete i just started with this process and extended it interpolated in this way and and i got discrepancies between the true evolution and its sort of most obvious Markov approximation, and those discrepancies were exactly the formula for interference, that was very surprising to me.

2:47:10And that had a big impression on me. I'll stop there for a moment because I know there's more to say, but I'm going to stop there for a second. So we're running out of time. We are. Yeah. And a lot of that went real quick. I'm puzzled by a couple of things, and I'll just say what I'm puzzled by. Sure, please. Yes. I mean, one is you say it's a two state system, but by the time you get to the screen, there better be a whole bunch of states that is their locations that they can land. Right. Or else I don't see what I'm going to see in terms of interference bans. So I'm a little puzzled. Let me make another comment.

2:47:47Feynman's argument in in his lectures is dreadful. It's just wrong. It's just wrong. He thinks from looking at statistics with one slit closed and the other open, and then looking at statistics from the other slit closed and this one open. And that and just probability theory will give you a prediction for both slits open. That's crazy. Probability theory won't tell you anything. As far as probability theory goes, the whole thing could blow up when you open both slits. Anything could happen. Probability theory doesn't tell you what will happen. He's making an assumption that if it goes through one slit, that the openness or closeness of the other slit can't make any difference.

2:48:34It must be irrelevant. That's false. And that's what we know from this experiment for sure is false. We put a phase shifter in one of the slits and we'll get different interference bands all the time. So, I mean, that argument that he gives there where he's suggesting, gosh, we have to go beyond classical probability. That's crazy. He was way off base. you're preaching to the to the choir tim totally agree oh you know i'm so i'm upset that we begin a lot of students first see quantum mechanics starting from this argument i think it has a lot of problems um okay so but so you're 100 i'm doing it with a two a two for simplicity two configuration you can do it with with a thousand in fact um uh i i've uh i was contacted by by uh someone who was very interested in this theory and he decided to do it for a system with way more ingredients and you begin to see what looks like the standard interference pattern show up.

2:49:28But I want to make now the extra thing because now we're seeing what looks like interference effects. And by the way, this is also reducing what interference is. This is now saying interference is a very different thing from what we've been saying it was. It's not a question of waves. Interference is just a discrepancy between evolution that's truly not Markovian and whatever convenient Markov approximation you might have wanted to make. I mean, Markov approximations are extremely useful. We model stochastic systems in nature all over the place that we have no good reason to think at the level of coarse graining we're working at should be treated as Markovs.

2:50:05People use Markov stochastic processes, Markov chains to model lots of phenomena that we know are only approximately true. And then there's a discrepancy because our Markov model is not going to get everything right. My claim is that those discrepancies, broadly speaking, are what interference is. And the interference of quantum mechanics is just one more example of that discrepancy between Markov and non-Markov. So this is all very suggestive, right? I mean, I'm not just proposing this. It just pops out and it's very exciting and very suggestive, but it gets even better. Now let's suppose that we make the experiment a little more complicated.

2:50:43let's suppose that we introduce an extra qubit that will serve as a detector okay and this is not going to be an external abstract primitive observer i'm going to model the whole thing as one big stochastic process so now i've got four configurations because i've got the two for my first qubit the actual traveling one and then two from the detector qubit and just it's two four possibilities all together and i'm not going to be able to go through the mathematical details and the time we have, but again, I can have you direct people to the description. But if you give the detector a very deterministic kind of dynamics, which we're modeling it how we want to model it, if we model this detector so that if our traveling qubit is in the upper region, then the deterministic detector stays in its original initial configuration, doesn't change.

2:51:37if the traveling qubit goes through the lower uh part of the apparatus the lower hole i guess then the uh detector qubit transitions deterministically it definitely transitions okay and and and this is obviously an idealization and you can drop the idealization but i'm going to idealize it just for present purposes so the the little detector is very deterministic in how it behaves you can model the whole thing as one giant staccato process the matrix gets big because it's now going to be a four by four matrix and you have to write everything out. It's not convenient, but the point here is not convenience.

2:52:09When you marginalize over the detector, classical marginalize, no quantum tracing, whatever. You just want to know, okay, what is the final probability distribution for the original traveling qubit going to be? The interference effects go away. They disappear. They exactly disappear. And if the interaction between the the detector qubit and the traveling qubit is not perfect, it's not ideal, then you won't completely lose the interference effects. You won't lose them partially. Now, why would we marginalize? Because realistically, a detector is going to be part of a much bigger macroscopic thing.

2:52:42We can't possibly keep track of everything. We marginalize the same way we marginalize in any classical ordinary probability story. And we lose the interference effects. And at the level of the stochastic law, we get what looks like a Markov event. The indivisible unistochastic matrix factorizes across the two times, and suddenly we get the generation, the spontaneous generation of a new division event that wasn't there before. It's not a global statement. We're not seeing the whole universe as a division event, but at a myriological level, for the myriological level of the original traveling qubit, it has this division event.

2:53:24And although we don't have time right now because we're just out of time, I've axiomatized this indivisible theory. And so you have to be very precise in the axioms how you deal with these spontaneously generated division events. But there's no assumption here that, and I know this is obviously a very delicate question, Tim, but I'm not going to make a metaphysical statement about this. This is just a math statement. But at a math statement, there's no time irreversibility or violation of time translation invariance at a global level, except maybe the very first initial to get the process started.

2:54:02But what you get is the spontaneous breaking of time translation invariance and time reversal invariance because these interactions are directed. The qubit, it evolves, the detector qubit evolves differently forward from how it evolves backward. And this means that the evolution is Markov to the past, but not to the future. And so there's a sense in which, you know, you're getting forward evolution that's definitively forward evolution from that moment. And so you're spontaneously generating this sort of error of time. I should just hasten to add, and this is just a nice, funny additional feature, is that unistochastic matrices, because they're doubly stochastic, have another very interesting property.

2:54:40They are entropy increasing. A doubly stochastic transition matrix always increases, or at least the entropy is always non-decreasing. So it maps on rather beautifully to, obviously more work is here to be done, but beautifully to a kind of sense in which we're getting something like the second law. So that's a lot. I apologize. There's a lot to say there. But my only point is that once we can do these sorts of experiments, you're like, oh, what other kinds of measuring apparatuses could I have? Could I modify the dynamics of my detector qubit? Just give it a difference to change the stochastic dynamics so that it could measure other things, things that are not beables, not the upward down configuration.

2:55:19The answer is yes, it can measure other things also. The things it's measuring in this case are not transparently reflecting the ontology of my traveling qubit, but as far as the detector is concerned, the detector gets a reading on them. And as far as the detector is concerned, it looks like it's just measured something. And so this can account for the contextuality of observables in quantum mechanics. We've got the Beobles, which in Hilbert space language form a commutative subalgebra of the full algebra of observables. And we also have these other non-commuting observables, what I call the emergibles, which Bohr described in 1935 as being emergent results of the interaction of a detector and the system being studied.

2:55:58And you get those two by just changing the dynamics of the measuring device. These measuring devices in this indivisible theory can measure lots of things, some of which are measurements in the literal sense that they're revealing some existing property. And in other senses, they're measurements in quotation marks the way that Bell said. We shouldn't call them measurements because they're not revealing something. They're just an experiment that produces some result, some probability. And I hasten to add that you can compute the probabilities with which the detector will get results from any of these things in the accord with the predictions of the Born rule.

2:56:27So this is obviously like, I mean, this was not, I didn't begin this project thinking that this was going to work. But every time I thought, oh, we're certainly not going to be able to have this show up or have that show up, stuff just shows up. So the picture of the world that I'm describing here is a picture in which there's just a configuration of the world, like the Shoemaker world. And the laws are indivisible. They're not Markovian. but they can be represented in a quasi-Markovian sense using a Hilbert space picture. The regalia, the appurtenances of a Hilbert space picture collectively make it into like a hidden Markov model with complex numbers and lots of arbitrariness and the definition of things.

2:57:12And that's kind of the picture of the world that's here. It's very simple to axiomatize. and in a sense it's kind of modest in terms of what it's demanding of nature um i'm not saying that one can't say what one can't propose that the additional ingredients of the hilbert space picture like wave functions or or quantum states i should say it can't be there i'm not making a an argument that one is a is obligated to to avoid reifying those things i'm just saying that from this point of view one could think of it as optional if one wants a somewhat more humble or modest ontology in the sense of only having local vehicles and not introducing more stuff, you might think you have to make the laws unbelievably complicated.

2:58:02Tim earlier said, if you remove the causal Markov condition, you get a proliferation of extremely complicated causal Markov models. You might have worried that would be the case. I was worried that would be the case if you drop the Markov condition. But it turns out that these indivisible processes are in some sense even simpler to formulate mathematically than Markov processes. They have fewer ingredients in them. When you actually write down the stochastic laws, you find that they're in some sense sparser and simpler than they are even for a Markov chain. And finally, get back to the question of how we get the causal Markov condition back again.

2:58:34If you've got a big macroscopic system, it's undergoing these division events all the time. they are the stochastic process analog through the stochastic quantum correspondence to what on the hilbert side we would call decoherence and either decoherence is getting rid of interference effects from macroscopic systems or it's not yeah and if it's not then quantum mechanics as we know it is wrong but if it is then on the stochastic process side is describing the generation of all these marco events and so for macroscopic systems we return to essentially a Markov-Process. A Markov-Process, the causal models you'll build out of these things.

2:59:13We don't have time to talk about causality, but the causal models would satisfy the causal Markov condition. And Ehrenfest's theorem will say that the averages, which are now bona fide averages of things happening, are going to evolve in a way that resembles classical equations of motion. So there's a lot more details to say that we didn't have time to talk about, but broadly speaking, that's the picture I'm actually presenting. You are now selling Shopidency to you, 43 Unidos or 14osen. With the A-Rezept이라고 version, you are Wang Experts to get your Desавлив medications very Augen. You just reject the especial average, a страни meant to get on your chest license withknowing and waste istem Twitch через 20 to 20€ means masterostaturated care, so, could you sell the product comparatively?

2:59:57At szofie, try the now version in the� человек. Okay. Well, I'm sure we've tried the patience of anybody who's still listening. Let me just say two words in response. Okay. For anybody else, a lot just happened. Yes. From the last time I read a word to now, you just made a lot of claims. Yeah. And we obviously don't have time to go through them. I'm still I have to say I'm still puzzled and I'll just state why. From the beginning, I said, gee, we've got these interference effects. This kind of interference we associate normally with waves, water waves, electromagnetic waves. I think there's probably something in the physical world that's a wave, that's a field-like thing that evolves by a wave equation.

3:00:50That would at least be an obvious way to account for that phenomenon. Now, you're telling me you can account for it without that by this. I can't absorb. There's just no conceivable way you could explain or I could absorb how any of that's supposed to work. And it's whereas in the beginning when you were just talking about these, you know, putting in these intermediate things I could follow just for anybody at home. If you couldn't follow, I have to say I couldn't follow. There's a lot of stuff there that would probably take a long time to work through. But that's, I think, just where we need to.

3:01:26Yeah. You know, that's entirely fair. I will refer you and anyone who's interested to the detailed calculations. There's a write-up on just the cubic case. There's a more recent paper that I can point you to where a more extensive calculation for more states was done. And then the general theory of how we get these division events to show up is in a paper that's published in the Journal of Philosophy of Physics just this calendar year. So, and some of those are technical. I mean, you have to do some work. It's not like this whole project is just trivial. Conceptually, like the axiomatic ingredients are simple.

3:02:07We just have configurations revolving indivisibly, and that's kind of the story. But to then show that this is empirically adequate and can save the phenomena, can account for the things we see, that's going to require work. And I'm not going to propose that we do quantum mechanics practically in this way. I mean, there's no question that you use, I mean, hidden Markov models are very convenient. I mean, there's a whole theory of epsilon machines that is very, very interesting that people can look up. but Crutchfield, you know, and so forth, developed that to handle non-Markovian systems as hidden Markov models, and everybody loves using these.

3:02:40They're really very powerful. They're very abstract, and we make them a hidden Markov model by introducing, like, fake extra stuff. And I should just say that now that we've, I've covered all of this, let me now quickly turn around and just say, when you have a stochastic process and you introduce an extra hidden or latent variable, and the language is a bit ironic because now the wave function or quantum state is the hidden variable, not the local variables. When you do that, in general for a hidden Markov model, this new variable will have an extraordinarily large number of configurations. This is another telltale feature that you've got, a hidden Markov model, which of course is all true of the wave function.

3:03:23But there are also some problems of uniqueness. And this is a question I wanted to ask you, Tim, because it's been bothering me a long time. so um and and the answer might be simple this is not a gotcha question i genuinely this is something i'm just very very confused about um in the bohm picture there's a pilot wave it's a wave function it's a complex function configuration space and it guides this equation guides the particles around There's a class of transformations called Foldy-Wauthausen transformations. transformations. I doubt most people have heard of them. They were introduced in 1950 in the context of the physics of spin-half particles, the Drac field.

3:04:11And they didn't really make their way into the philosophy literature until Harvey Brown, I think, independently re-derived them in a paper in 1999 called Aspects of Objectivity in Quantum Mechanics. It's on the first page. You can just see it right there in that paper. it's a class of transformations that change the quantum state or the wave function. Now, on the Bohm theory, the local beables are particle positions, and Bell himself, and I don't have to check exactly what Bohm exactly said, but Bell said this, argued that all the empirical output of the Bohm theory supervened on the distribution of positions of particles.

3:05:02Now, that's fine. There's no in principle issue with that, except that these transformations you can do, there's a restricted class of them, the class that commutes with the position operator. and these transformations mangle up the wave function but don't touch the mod square of the wave function don't touch the positions at all and they alter the guiding equation they alter the trajectories and there's no canonical frame to use in the sense that like there's no preferred like you can do one of these full devout housing these restricted full devout housing transformations what they effectively do is they change the phase factor of the schrodinger wave function by a space-time dependent phase and this doesn't change any of the mod squaring it doesn't change the probabilities of the overall i should say the overall wave function it's not you know the whole thing but it does modify the guiding equation and it changes the trajectories the particle takes but because by assumption or by argument, I mean, the question is how you rigorously show this, that all of the empirical output supervenes on just the positions of the particles or their distribution.

3:06:25This troubles me because it seems like the guiding equation and the trajectories of the Bohm theory are radically unfixed. And I'm wondering if you can maybe help me understand why I should not be worried about this problem. I mean, I think I could. um i i'm sure robinson is like wants to gnaw his foot off at this point about another thing let me try in one minute to say uh look everybody who works on the pirate wave theory knows you can actually change the guidance equation in certain ways that won't change the statistics for the outcomes okay that's well known often those changes look just crazy and arbitrary and you You wouldn't do them because they just look kind of loopy.

3:07:08And we need to make a distinction between gauge transformations, active transformations, passive transformations. It's clear you can pick any mathematical object, screw around with it in some way, screw around with the dynamics in an opposite way, and end up with the same empirical predictions. So, you know, the question is, if the wave function represents a real physical thing, what are its merely gauge degrees of freedom? What are its physical degrees of freedom? That's not an easy question. You can answer it like the overall phase. Does that correspond to a physical degree of freedom? Or is it just an artifact of the representation?

3:07:45I don't think there's an algorithm to answer that. I think you have to look very carefully at the individual cases. Often normal things like you're doing something really screwy there for no purpose. We'll make one look a hell of a lot more plausible than the other, even if you say, gosh, but the appearances would be the same. But I think these would be very detailed questions you'd have to go through, and there's not a one-size-fit-all, even algorithm to try and sort through it. Yeah, so the reason I just bring this up is these fully Vauthausen transformations are very simple. They look just like electromagnetic gauge transformations.

3:08:23No, I would think they sound like local gauge transformations of electromagnetism. Yeah, except that the gauge connection is the Hamiltonian, it's this weird thing. But yeah, so one, The argument, I mean, you can make the phase of the universal wave function on the bone picture. You can change its phase at every space-time point to any other phase you want, which makes it very hard to think that the phase at all, at any point, is somehow physical or not engaging. It's a local gauge transformation. We're kind of familiar with that stuff. But anyway, I think this would take us wait. This would take us a long time, yeah, yeah.

3:08:57But this worries me. It's one of the reasons, one of the things that has made me skeptical that the pilot wave could be physical, but it's not an open and shut case. Make clear I'm not making any open and shut cases here. mm-hmm well i wish that within the span of three hours it were ever possible to really resolve uh a substantive issue in in either philosophy or or physics but i'm really glad that we got to have this or i got to host this conversation on on my on my show and i hope that it continues either on mine at some point or on another show and i'm sure there's there's plenty that a lot of people would love to hear the two of you talk about.

3:09:40But for now, thanks again so much for joining me for this. It's a real delight. It's lovely to see you both. Be well.